msp.org · 1BDJGJD +PVSOBM PG.BUIFNBUJDT INTHISISSUE— Richard Horace Battin, Note on the “Evaluation of an integral occurring in servomechanism theory ...

PacificJournal ofMathematics

IN THIS ISSUE—Richard Horace Battin, Note on the “Evaluation of an integral

occurring in servomechanism theory” . . . . . . . . . . . . . . . . . . . . . . . . 481Frank Herbert Brownell, III, An extension of Weyl’s asymptotic law

for eigenvalues . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 483Wilbur Eugene Deskins, On the homomorphisms of an algebra onto

Frobenius algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 501James Michael Gardner Fell, The measure ring for a cube of

arbitrary dimension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 513Harley M. Flanders, The norm function of an algebraic field

extension. II . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 519Dieter Gaier, On the change of index for summable series . . . . . . . . . . . 529Marshall Hall and Lowell J. Paige, Complete mappings of finite

groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 541Moses Richardson, Relativization and extension of solutions of

irreflexive relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 551Peter Scherk, An inequality for sets of integers . . . . . . . . . . . . . . . . . . . . . 585W. R. Scott, On infinite groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 589A. Seidenberg, On homogeneous linear differential equations with

arbitrary constant coefficients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 599Victor Lenard Shapiro, Cantor-type uniqueness of multiple

trigonometric integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 607Leonard Tornheim, Minimal basis and inessential discriminant

divisors for a cubic field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 623Helmut Wielandt, On eigenvalues of sums of normal matrices . . . . . . . 633

Vol. 5, No. 4 December, 1955

PACIFIC JOURNAL OF MATHEMATICS

EDITORS

H.L. ROYDEN

Stanford UniversityStanford, California

E. HEWITT

University of WashingtonSeattle 5, Washington

R . P . DILWORTH

California Institute of TechnologyPasadena 4, California

* Alfred Horn

University of CaliforniaLos Angeles 24, California

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HERBERT FEDERER

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P.R. HALMOS

HEINZ HOPF

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GEORGE POLYA

L J . STOKER

KOSAKU YOSIDA

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CALIFORNIA INSTITUTE OF TECHNOLOGY

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COPYRIGHT 1955 BY PACIFIC JOURNAL OF MATHEMATICS

NOTE ON THE "EVALUATION OF AN INTEGRAL OCCURRINGIN SERVOMACHANISM THEORY"

R. H. B A T T I N

In a recent paper [1] W. A. Mersman considers the evaluation of the integral

>=— Γ -2πi J-ooi hix

dx

where gix) and hix) are polynomials in x of order 2n — 2 and rc, respectively.

Because of the importance of Mersman's result the present writer wishes to call

attention to an alternate and somewhat more direct evaluation of this integral.

We shall utilize Mersman's notation in the main and begin with his equation

(3). By division it is clear that

where it is important to observe that each of the quantities Bj^ will, in general,

depend upon k except the first which is simply B^^ - α o Then

X-Xk -x ~xk

T T.s=0 7=1

n 2r

Σ, Σ2(n-r)

In the above expression it is understood that as = 0 for s < 0 or s > n and

Bjfr ~ 0 for j > n. Mersman's equation (3) then becomes

Mn-r)

r = i

n 2r

/c=i

Received September 20, 1953.

Pacific J. Math. 5 (1955), 481-482

481

482 R. H. BATTIN

For simplicity we define

Ak Bjk J = If 2, , n

so that Fι -I. There results the following set of n linear algebraic equations:

2Γ σ

/=2 2 α °

Using Cramer's rule we may now solve directly for / to obtain Mersman's result

as expressed by his equation (6) .

R E F E R E N C E

1. W.A. Mersman, Evaluation of an integral occurring in servomechanism theory,Pacific J. Math. 2 (1952), 627-632.

MASSACHUSETTS INSTITUTE OF TECHNOLOGY

AN EXTENSION OF WEYL'S ASYMPTOTIC LAW FOR EIGENVALUES

F. H. B R O W N E L L

1. Introduction. Let D be a bounded, open, connected subset of the plane

E2 whose boundary B = D — D is a simple closed curve whose curvature exists

everywhere and is continuous with respect to arc length; consider the eigen-

values λ - λn > 0 of the problem

(1.1) V 2

α + λ u = 0 on D, u = 0 on B,

where u(x) is to be continuous over ϋ and have continuous second partials

over D, ^ being the Laplacian. It has long been known (see [7, bibli-

ography]) that in this situation, with 0 < λ^ < λ^+i repeated according to

multiplicity, the asymptotic distribution of λn is given by Weyl's law

μ2(D)( 1 . 2 ) / V U ) = T 1 = — t + o ( t ) , t—» + oo,

where μ 2 ( D ) is the two dimensional Lebesgue measure of D. This can be ob-

tained by Tauberian theorems from the estimate

~ 1 μ a ( D ) lnω C

d 3 ) Σ TTT ,=~Λ + - + O ( ω - 5 / 4 ) , ω - ^ + co,n=ι An(λn + ω) 4 77 ω ω

(see Carleman [ 2 ] for the E3 analogue). By domain comparison methods [3,

p. 386 ] Courant has shown that o(t) in (1.2) can be replaced by O(\Jt l n ί ) .

In a recent paper [6, p. 177, equation 16] Pleijel replaces the estimate (1.3)

by the very much stronger

(14) y l n ω , c nB) 1 ! o l 1 )

over ω > 1 in case the curve B is very smooth (that is, it has an infinitely

Received October 19, 1953.Pacific h Math. 5 (1955), 483-499

483

484 F.H. BROWNELL

dίfferentiable parametric representation), where C is an unknown real con-

stant and KB) is the total length of S. Pleijel's estimate (1.4) follows easily

from a deep investigation jointly with M. T. Ganelius on the compensating part

of the Green function, as yet unpublished. This investigation uses integral

equations over the boundary B, while estimates like (1.3) come from a simple

application of the maximum principle over D to the modified Green function.

Pleijel suggests it should be possible to sharpen (1.2) by using his methods to

investigate the analogue of (1.4) over complex ω.

It is the purpose of this paper to show that from (1.4) alone we can replace

(1.2) by

(1.5) ΛK)4 77

in a certain sense. Precisely our result (2.13) is that with

KB)1 4ιr 4π J

have

over all real u >_ e and all p > 0 for some M < + oo. Moreover, if Nit) has an

ordinary asymptotic series in powers of ί, it must be consistent with (1.5). We

discuss briefly the possibility of sharpening (1.5) by replacing averaged 0

estimates by ordinary ones. We also note the utility of our consistency result

in proving false a conjecture of Minakshisundaram [5, p. 331, no. 2] about the

asymptotic behavior of Nit).

Clearly our theorems will apply to give results like (1.5) for a wide variety

of more general problems than (1.1) for which estimates like (1.4) obtain; in

particular such results hold for (1.1) in 3-space £3 .

2. Results and proofs. The difficulty arising in trying to get an asymptotic

series like (1.5), with 0 replaced by an ordinary 0 or o, is that Tauberian

theorems yielding such results seem to require essential nonnegative condi-

tions after subtracting all but the last term of the series. It is quite clear that

Nit) does not satisfy such a condition. For this reason we use an indirect

AN EXTENSION OF WEYL'S ASYMPTOTIC LAW FOR EIGENVALUES 4 8 5

Abelian type argument [4, p, 224] to get averaged error estimates of the 0 type.

The two first theorems here establish the significance of these averaged

error estimates, which despite the resemblance to Gaussian summability seem

to be little used for asymptotic series. Crame'r [ 1 , p. 819 and p. 823, ( 3 ) ] has

used Caesaro-1 type averaged error estimates on lattice point problems, but

such processes do not appear strong enough for use here.

Throughout the paper all integrals are to be understood in the Lebesgue or

Lebesgue-Stieltjes sense, and for the following two theorems it is understood

that Fit) is to be real valued of bounded variation over every finite interval

of [0, oo), with positive b a continuity point of Fit). Also \dF it) | stands for

dVpit) where Vpit) is the total variation of F over [bf t].

T H E O R E M 1. //

[°° i r ° \ d F i t ) \ < + o oJ b

for some r0 > 0, if

φ(s)= f° fs dF(t),

which must exist and be analytic in s over H[s] > ΓQ, also has an analytic

continuation without singularities throughout R [s ] > 0, and if

over 0 < r <^ro and all real v for some Mi < + oo and h > 0, then over all real

u >_ e and p > 0 we have

(2.1) I Γ β-<pV»> (In <•/«»» dF{t) < l2Mt exp ( l + Plh* + —\Jb \ \ 2 2p

In v i e w of ( 2 . 1 ) i t b e c o m e s c o n v e n i e n t t o de f ine Fit) = 0 if it)) over

t >^ b for s o m e n o n n e g a t i v e f it) d e f i n e d over t >^ k > 0 if for e a c h p > 0 t h e r e

e x i s t s s o m e Mp < + oo s u c h t h a t the left s i d e of ( 2 . 1 ) e x i s t s and i s <_Up f iu)

for a l l u >_k. With t h i s d e f i n i t i o n we c a n r e s t a t e t h e c o n c l u s i o n of T h e o r e m 1 a s

Fit) =

486 F. H. BROWNELL

over t >_ b with k = e. Note that in (2.1) Up —> + 00 as either p —» 0 + or

p—»+oo, so that (2.1) becomes meaningless then. The significance of the

result (2.1) is greatly increased by the following consistency theorem.

THEOREM 2. //

f t'r°\dF(t)\ < + o ob

for some r0 > 0, if

over t >_ b, and if

as t —> + ex) for some rλ > 0, then C\ — 0.

Proof of Theorem 1. Let

4 ( y ) = exp \~ — y2 ~zy]

for p > 0 and any complex z; thus

vNow

M> [Tt'r° \dF(t)\ >r'Γ°VF(r)

Jb

shows

VF(t)=O(tr°);

thus

Jy=ln b z


exists as an entire function of z over all real ω and all complex z The Fubini

theorem also shows g(z9 ω) G Lt(-00,00) over ω with

(2.2) f°° ( ( \= -exV( (v + - \ )ψ(z+iυ)P \ 2p2 \ i l l2πJ'°° P \ 2p

over H[z+iv]-H[z]>_ro9 v being real. But the right side of (2 .2) is in

L]i— oo, oo) over v since

fr \dF(t)\,b

and thus the Fourier transform inverse yields

( 2 . 3 ) [ ° ° f ( ω ~J=\ b[ f (y=\n b

1 Γ~ I (v + z / ί ) 2 \ , , ,„,, dv

= exp \φ (2 +iv)eιvω —

v^J-~ \ 2p2 j P2P

for R [ z ] >, r 0 . T h e g i v e n e s t i m a t e on φ{s) a c t u a l l y m a k e s t h e far r i g h t s i d e

of ( 2 . 3 ) e x i s t a n d be a n a l y t i c in z t h r o u g h o u t R [ z ] > 0, and t h u s by a n a l y t i c

c o n t i n u a t i o n ( 2 . 3 ) h o l d s t h e r e a l s o . T h u s wi th z = r w e h a v e for e v e r y p o s i t i v e

r and p a n d for e v e r y r e a l ω the e s t i m a t e

I Γ°° / P2

(2.4) exp - — ( ω - -

I Jr=ln6 \ 2

2p2

Multiplying (2.4) by e r α )and letting r = 1/ω > 0 we note for ω > 1 that

488 F. H. BROWNELL

r2 1 1rω H — In r = 1 + + In ω < 1 + + In ω,

2p2 2p2ω2 ~ 2p2

thus with y = In t and ω — \nu >^ 1 we get the estimate (2.1) as desired.

Proof of Theorem 2. As before we have

\ F ( t ) \ < \ F ( b ) \ + VF(t) = O ( t r ° ) ,

so that we can integrate by parts in the left side of (2.1) and obtain from

F ( t) = 0 (In t) over t > b the estimate

(2.5) ply'lnb

over ω >_ k > 0. Now we are given

C l ί Γ l + / ( ί ) ί Γ l

over ί > b with l i m ^ + o o / ( ί ) = 0 Thus multiplying (2.5) by e ι , letting

y = ω — Λ;, and taking ω —> + oo we get

f Γω-lnfe / p2 \0 = lim j cι I Λ expj-ΓiA; — x \

α>—+ oo I J-°° \ 2 /

2 \\dx

Defining f ( t) = 0 for t +Oo / ( ί ) = 0 , and thus also

limω_» + o o / ( eω"x) = 0 , dominated convergence applied to (2.6) yields


= Ci I x expl ~rγ% x )dx.J -βo \ 2 /

But

< 0

for rι > 0, so that c^ = 0 follows.

To apply these two theorems we use a standard contour integral transforma-

tion on Pleijel's estimate (1.4). The contour Cp, p >_ 0, in the z plane is de-

fined to be first along the negative real axis from — oc to — p, then around the

circle z = peι from θ = - π to θ = 77, then back along the axis to -en. On this

contour we define

with θ ~ — π9 — π < θ < π, θ = π on the three parts respectively. The well known

results are formulated in the following two lemmas (Carleman [2]) , and we

sketch the proofs for the sake of completeness.

LEMMA 3. IfO< λn < λ n + l f an real, if

n

and if

λ 2

then

converges absolutely and is analytic in all complex z except for simple poles

at each λn. Moreover, for 0 2,

2 * an An

converges absolutely and uniformly over ft[s] ^ 2, and over R [ s ] > 2 we ob-

tain

(2.7) r - / h(z)(z) S - l

LEMMA 4. //ίAe assumptions of Lemma 3 are satisfied and if

(2.8) Λ(~ω)= ς w p

ω V . ' P

l n ω + Q ( ί )over ω > 1 with 0 < r^ < rjc.ι < < rγ < 2, then

an analytic extension into ft [s ] > 0 e#cepί /or poZes at rp,

(2.9)in ίr(rD — 1) 1s i n

s i n+ ton

(s-rp)2

cos 7r(r p -l)

analytic in s throughout ft [s ] > 0, aτιcf

Λ )\ 77^

over 0 < r < 2 ancί all real v for some M2 < + ω .


We remark that TΊ < 2 is no real restriction in (2.8), since the assumptions

of Lemma 3 imply \Ίmω _ + oo h ( - ω ) = 0 . In demonstrating Lemma 3, the stated

analyticity of h (z) is clear as well as

; l/ \ n = 1

SO

g ( s ) = _ L / h(z) J t2 π ί CP (z)

exists and is analytic in s over R [s ] > 2. To show

there for (2.7), let Cm be the vertical line contour from xm — ΪOO to xm + ico for

%m with λnml < xm < λnf so that using the estimate on h ( z ) to shift from

Cp to Cm we obtain

(2.10) g(s)~ T a, λ : s = — f h{z) —

for R [ s ] > 2, h(z) having the residue

at λ.

To pass from (2 .10) to ( 2 . 7 ) , note that

t lim sup λ2

n ( λn - λnm i )f] = + <

since otherwise

492 F. H. BROWNELL

would be bounded by

which contradicts λ π —» + oo and therefore contradicts

n ι n

Thus there exists a sequence nm such that

nm < Λm +19 λn — λn_ i > 0, and λ^ ( λn - λn. t ) —> + oo as m —» + oo for

We choose

for n = nm, so that

K 1_ ^ 1 + > i

and

< Λm + — = *m + 2 * J

n

With z = xm + it and s = r + ίvf r > 2, clearly

and

I 1 I e x p U | t ; | / 2 )

with L (v) = M exp ( π \ v |/2) make


k(z).s-1

\ L(v)over \t\ < λ n - xm ,

\h(z) < (χm)2'r L(v)o v e r λ n - x m <\t\ <xm9

and

\h(z)s-i

K \ L(v)I K \ L(v)over χm < \t I.

Thus integrating over these respective parts of Cm, and using

the right side of (2 .10) —> 0 as m —> + oo and ( 2 . 7 ) follows.

P a s s i n g to Lemma 4, from the estimate ( 2 . 8 ) it is clear that

dzh(z)

e x t e n d s a n a l y t i c a l l y f r o m R [ s ] > 2 t o R [ s ] > r i A l s o f o r rι < R [ s ] < 2 ,

C ^ c a n b e s h i f t e d t o C o y i e l d i n g

(2.11) g U ) =

Now here

sin 77 ( s - 1)

77

h(z)dz sin π(s - 1) /*oo dcύ

/ h(-ω) .Jo „ s-i

ω

ω s i n τ 7 ( s - l )=

s-ι _ τ r ( s - 2 )j A ( - l ) + / h (-ω) ,

which is analytic in s over R [ s ] < 3, having a removable singularity at s - 2.

Also

in 77 ( s — 1 )s i n Γ-2in 77 ( s - 1 )sin

ωs -1 77 ( s - r )

and

s i n 77(s — 1 )

/

oo c?ω s i n 77 ( s •ω In ω =

ωs-ι π { s - r

c?ω s i n π(s - 1 )

494 F. H. BROWNELL

with principal parts

sin 77 (r - 1)

π(s - r )

s i n 7 7 - ( r - l ) c o s τ r ( r - l )and +

π(s-r)2 s-r

respectively at s =r. Thus (2.9) clearly follows from (2.8) and (2.11). Also

from

i n 7 7 ( 5 - 1 ) 1 < 2eπ\v\ a n ds in/

oo 1 day 1

2 r-1 Γω ω

the stated estimate for gîs ) follows.

We combine Lemma 4 with our two previous theorems to obtain the following

result.

THEOREM 5. If the assumptions of Lemma 4 are satisfied with

lp sin (πrp) = 0

in (2.8), then

- Σ an

satisfies

( 2 . 1 2 ) H(t) = \ •' — lp c o s π rp) J + O ( l n ί ) ,

over t >_b where 0 < b < λ 1 # Furthermore, if H(t) has an ordinary asymptotic

series in powers of t as t—» + oo, such a series must coincide term for term

as far as it goes with the terms o / ( 2 . 1 2 ) .

Proof. Let

F{t)=H(t)~ -= l ' P

p lp COS TΓΓp) I ,77 / I

and note that


ir-s

t'-'-i dt = —s ~r

for R [s ] > r and b > 0. Also with 0 _2. Thus from Lemma 4 we see that

t's dF(t)b

has an analytic continuation without singularities into R [ s ] > 0 by the can-

cellation of principal parts at each rp = s. Also the conditions of Theorem 1

are satisfied with r0 = 2 and h = π; thus (2.1) yields (2.12). Theorem 2 gives

the consistency statement obviously.

To apply this theorem to our problem (1.1), we remark that the desired

condition

follows from Green's function being in L2(D x D), and thus a Hilbert-Schmidt

kernel. Thus Pleijel's estimate (1.4) yields (2.12) with

2

k = 2, r! = 1, mi = C, lχ = , sin ( πr±) - 0, cos ( nr±) = - 1,4 π

1 Z ( B ) . .ro — — 9 rrio — , lo — 0> s i n v ^ Γ 2 ) = 1 >

2 8

and we can state the following.

COROLLARY 6. Let the open, bounded, connected set D in the plane E2

have its boundary B an infinitely differentiable Jordan curve so that Pleijel's

estimate (1.4) holds for the problem (1.1). Then over t >_ λj2 we have

2

(2.13) /VU)= Y, 1 = tλn<t *π

496 F. H. BROWNELL

and as in Theorem 5 any ordinary asymptotic series for N ( t ) must be consistent

joith (2.13).

If we consider the real valued eigenfunction un(x) of problem (1.1), in

place of (1.4) Pleijel gets [6, equation 6 and second equation of p. 177] over

X G D and ω > 1

- \un(x)\2 1 lnω C(x) 1(2.14) V — — = + ' +

Tί λπ(λπ + ω) 4π ω ω 2π

/ - 2 Λ r ( x ) V u Γ \

•°h?H/4 > 0, r ( x ) the distance from x G D to β, the 0 symbol being uniform over

x G D as well as ω > 1, Now K 0 ( Γ )> t n e modified Bessel function of the second

kind and zero order, has

as r — » +00 [ 8 , p . 374] . Thus for each fixed x G D , with r ( x ) > 0, we have

over ω >^ 1

, r N M ) | 2 1 lnω C(x)(2.15) ^ ΓΠ "ϊ β 1 + + K

^ ί λ n (λ π + ω) 4τr ω ω \ω

1\

ω

2}

where the symbol 0χ now depends on x G D. It is also easy to see that at each

x ^ y with x, y G D we have over ω >_ 1

~ ^(xWy) C(x,y)

(2.16) Σ T T T x- — — +0 x ,/ 1 \

, y _ ,\ω2/

and indeed much better estimates than O(l/ω 2 ) hold in (2.15) and (2.16).

Also

+ c o

π=ι λ2

n

is known at each x G D; thus Theorem 5 yields the following.

AN EXTENSION OF WEYL'S ASYMPTOTIC LAW FOR EIGENVALUES 497

COROLLARY 7. Let D be as in Corollary 6, so that (2.15) and (2.16) hold

at each x ^ y with x, y £ Z). Then over t >_ λχ/2

( 2 . 1 7 ) £ K ( x ) | 2 = — ί + O ( l n ί ) , Σ, unix)uniy)=Oilnt),4

λn<t -*π λn<t

with consistency of these series with ordinary asymptotic series, if any, as in

Theorem 5.

3. Discussion of results. It is quite clear that O ( l n ί ) in (2.17) can be

replaced by much stronger estimates in the 0 sense, say O ( l / ί ) , since much

more than Oil/ω2) holds in (2.15) and (2.16). In (2.13) additional terms

enter if a stronger 0 type error estimate is required. These are due to additional

terms entering PleijeΓs equation (1.4), one of them involving the mean square

curvature of B, if 0 ( 1/ω2) is replaced by a stronger estimate.

A much more difficult and interesting question is the extent to which the

averaged 0 estimates in our results may be replaced by ordinary 0 estimates

for the problem (1.1). It is clear that by improving the Oieπ\v\) estimate on

the analytic continuation of

oo

Λ^ , s = r + ιv,

we can replace the Gauss kernel

I p2( A

in our definition of 0 by less well behaved ones. We could get ordinary 0 esti-

mates if we could use the characteristic function kernel X[_t ^ ( ω - y ) , but

since its Fourier transform is essentially v~l sin v, the analogue of the proof of

Theorem 1 would then seem to require stronger conditions on

£-^ n

than can be expected to hold.

It is known from the refined results of geometric number theory [1, p. 823]

that Mix), defined as the number of integer lattice points im,n) in the plane

satisfying m2 + n2 <, x, satisfies

498 F. H. BROWNELL

πx + O(xι/3).

Since

λ = U 2 + m 2 ) — , n > 0, m > 0

for the eigenvalues of (1.1) with D a square of side 6, the eigenfunctions being

products of sine functions, we clearly see that

/ V ( t ) = - IM — -b2 4b r- , 1 / 3 χ

= t y/T + O ( ί ι / 3 )4π 4 7r

for square D, 4[byt/π] + l being the number of lattice points on the axes.

This asymptotic result for Nit) agrees with (2.13), although the corners of a

square prevent it from satisfying the smooth boundary conditions required in

Corollary 6. By carelessly dropping the y ί term in going from Mix) to Nit)9

Minakshisundaram [5, p. 331, no. 2] is led to the conjecture that domain com-

parison methods [ 3 , p. 386] should yield

t + Oitί/3)4/7

for general domains D. Clearly the consistency statement of Corollary 6 makes

such asymptotic behavior impossible for Nit).

REFERENCES

1. H. Bohr and H. Cramer, Die neuere Entwicklung der analytischen Zahlentheorie,Encykl. der Math. Wiss., 2, part 3, no. 8.

2. T. Carleman, Proprietes asymptotics des functions fondamentales des mem-branes vibrantes, Forhand. 8th Skand. Mat. Kongress, (1934), 34-44.

3. R. Courant and D. Hubert, Methoden der mathematischen Physik, vol. 1, Springer,Berlin, 1931.

4. G. Doetsch, Laplace trans forms, Springer, Berlin.

5. S. Minakshisundaram, Lattice point and eigenvalue problems, Symposium onSpectral Theory, Stillwater, Okla., (1951), 325-332.

6. A. Pleijel, Sur les valeurs et les functions propres des membranes vibrantes,Comm. Sem. Math. Univ. Lund (Medd. Lunds Univ. Mat. Sem.), suppl. (Riesz) vol.,(1952), 173-179.

AN EXTENSION OF WEYL'S ASYMPTOTIC LAW FOR EIGENVALUES 499

7. H. Weyl, Ramifications of the eigenvalue problem, Bull. Amer. Math. Soc, 56(1950), 115-139.

8. E.T. Whittaker and G. N. Watson, A course of modern analysis, Cambridge, 1946.

UNIVERSITY OF WASHINGTON AND

INSTITUTE FOR ADVANCED STUDY

ON THE HOMOMORPHISMS OF AN ALGEBRA

ONTO FROBENIUS ALGEBRAS

W. E. D E S K Ϊ N S

1. Introduction. A linear associative algebra possessing a nonsingular

parastrophic matrix is known as a Frobenius algebra after the mathematician

who first investigated the properties of such an algebra [3] . In more recent

years the properties of this class of algebras have been studied in papers by

a number of mathematicians, notably R. Brauer, C. C. MacDuffee, T. Nakayama,

and C. Nesbitt (see References).

Since Frobenius algebras are defined in terms of the parastrophic matrices,

a natural question to ask is the following: Does a parastrophic matrix of rank

m of an algebra U of order n determine in some manner a homomorphism of Cl

onto a Frobenius algebra of order m? As the answer to this query is, in general,

negative, it is the purpose of this paper to investigate the question: When does

a parastrophic matrix of rank m determine in some manner a homomorphism of Cl

onto a Frobenius algebra of order m? First a "manner of determination" is

selected. Since the parastrophic matrices of G form a double CUmodule, various

ideals of & of annihilating elements correspond to each parastrophic matrix.

These are studied and conditions are developed (Theorem 9) which insure the

determination from these annihilators an ideal B such that the difference algebra

d - B is a Frobenius algebra of order m. These requirements are shown to be

necessary, also, in the sense that any homomorphism of Cl onto a Frobenius

algebra of order m implies the existence of a parastrophic matrix Q of rank m

which satisfies these conditions. Furthermore, the kernel of the homomorphism

will be the ideal B determined from among those elements which annihilate Q

as an element of a double CUmodule.

Basic terminology is introduced in v 2, parastrophic modules are defined,

and the order of such a module is discussed. In § 3 one-sided ideals determined

by the parastrophic matrices are considered, while § 4 is devoted to a study of

two-sided ideals determined by certain parastrophic matrices and of the homo-

morphisms of an algebra onto Frobenius algebras. Certain of the ideals

Received February 10, 1954. Presented to the American Mathematical Society May1, 1954.

Pacific J. Math. 5 (1955), 501-511501

502 W. E. DESKINS

introduced in § 4 have radical-like properties, and these ideals are considered

in § 5. A supplementary re 'c on the order of the radical of a Frobenius algebra

is given in §6.

The author wishes to express his gratitude to Professor C. C. MacDuffee for

his counsel during the preparation of this paper.

2. Preliminary remarks. Let U be a linear associative algebra of order n

over the field 3, and let βi, « f e n be an 3-basis for G. Multiplication in &

follows from the multiplication of the basis elements,

ei ej = Σ cijk ek i» ί = !> > n,k

where the Cijk are elements of 3, the constants of multiplication.

The associativity condition, written in terms of these constants of multi-

plication, is equivalent to each of the following sets of n2 matric equations:

(1) Q/Ri-Σ, cikjQk,k

(2) SiQj = Σ, ckijQk i f / * 1 . ' • • » Λ ,

k

where the matrices R(9 S(9 and Qι are defined as (c ; S Γ ) , ( c Γ ι s ) and (c S Γ ^), re-

spectively, where r denotes the row and s the column index.

Let a G G; then

a = a ί e i + ••• + a n e n ,

where the a( are field elements. Let

S(a) =atSi + + α n S π ,

+ ••• + anQn.

R{a) is called the first matrix, S(a) the second matrix, and Q(a) the para-

strophic matrix, of α. (Note that Q(a) as defined here is the transpose of the

parastrophic matrix as defined by other authors). The set R(®) of all the first

(second) matrices of 0* form an algebra which is a homomorphic image of U. The

HOMOMORPHISMS OF AN ALGEBRA 5 0 3

set o2 of all the parastrophic matrices of CL does not in general have this proper-

ty, but if the following definitions are made,

then 12 is a double Ci-module, the parastrophic module of G.

If a change of basis is made for U, the elements of R and & undergo similar-

ity transformations, while the elements of 2 undergo congruency transformations

[ 8 ] . Hence rank and symmetry are invariant set properties of 2 .

MacDuffee has obtained [7] necessary and sufficient conditions that R and

& be algebras isomorphic with U. A corresponding result for 2 is given by

THEOREM 1. 2 is of order m (as an Cί-module) if and only if the following

conditions are satisfied:

( i ) U contains an ideal w of order m such that the difference algebra

(1 — w is a zero algebra.

( i i) & contains no ideal of lower order with this property.

The proof of this theorem is a standard reversible procedure involving a

change of basis for U. Let there be n — m linearly independent linear relations

among the Q^; then there exist n - m linearly independent row vectors

Ti = (ti i, , tin ) i = m + 1, , n ,

such that

23 HkQk =° i> m.k

If B is a nonsingular n by n matrix with the Γj as its last n - m rows, and if

u i, , un form a new basis for Q,

Σ Pkiekk

where P = (pΓ ) = S"1, then the Qι are transformed into

i > m.

504 W. E. DESKINS

Thus if the new constants of multiplication are cf.^ , then

c.' k = 0 i% j = 1, . . , n k > m,

and uχ9 9um span an ideal ID of order m such that U — IΛ is a zero algebra.

The process is clearly reversible.

COROLLARY. // & has either a left or right identity element, then 2 is of

order n.

u is said to be a Frobenius algebra if °2 contains a nonsingular element.

THEOREM 2. U is a Frobenius algebra if and only if 2 is a cyclic module

of order n.

If 2 c o n t a i n s a n o n s i n g u l a r e l e m e n t Q, t h e n ( 1 ) and ( 2 ) imply t h a t

and 2 is of order ra since a Frobenius algebra possesses an identity element.

Conversely, if 2 is generated by an element Q and is of order n, then (1),

(2), and Theorem 1 imply that Q is nonsingular.

3. Ideals of CL Let 6 be a right ideal of Cl of order n - m. If a basis is

selected for Cl such that the last n - m elements of the basis span 6, then the

m matrices Q\, ,Qm have all zeros in their last n - m columns. The task of

determining a right ideal by a process involving reduction of certain elements of

2 through changes of bases of d seems formidable, if possible. However, a

somewhat similar process is given by the following theorem.

THEOREM 3. A parastrophic matrix Q of rank m determines a right ideal

6 of order greater than or equal to n - m.

Let B be the set of all elements b G Cl such that Q * b = 0. Clearly 6 is a

right ideal. That its order is at least n - m will follow from the next theorem.

That B may actually be of order greater than n - m is proved by the follow-

ing example. Let G have basis elements e\ and β2, ^\-^\^2 = e 2 e l = 0>

e£ = βi Then the corresponding (^ has rank 1 but B = CL

A more desirable result is contained in the following.

THEOREM 4. A parastrophic matrix of rank m determines a right ideal of

order n - m.

HOMOMORPHISMS OF AN ALGEBRA 5 0 5

Let Q be a parastrophic matrix of rank m. Then there exists a nonsingular n

by n matrix P such that the elements of the last n — m columns of QP are all

zeros. Let P effect a change of basis for A; that is, if P = (prs \ l e t

ui =Pkiek

be a new basis for U. If Q — Q(a), then Q'(a)9 with respect to the new basis,

is P QP and hence has nothing but zeros in its last n - m columns.

Now assume Q is of this form. Then

= Σ aiQi =ί Σ, aicsri\i * i '

so that

(3) Σ, ai cjki = ° 7 > m* k = 1, *, n.i

From (1) and (3) it follows that

(4) QRi= Σ

Hence Q * e{ ~ 0 for i > m, and B = (e m + j_, , en) is the right ideal determined

hyQ.

A right ideal of fl which may be determined in this way will be called a

parastrophic right ideal.

THEOREM 5. A sufficient condition that the ideal of Theorem 3 be a para-

strophic right ideal is that 12 be of order n.

Suppose Id = ( e m + 1 , , en) is determined from Q as above, and considereit i <. 77i. If @ * βj =s 0, then if =2 is of order nf (4) implies

which is impossible since Q is assumed to be of rank m.

Let Q = Σ,aιQι be in the reduced form described above.

506 W. E. DESKINS

THEOREM 6. If U has a right identity element, then aι = 0 for i > m.

Since U has a right identity element there are field elements f. such that

Then ( 3 ) implies

j > m.

The results of this section are obviously valid if the word "r ight" is re-

placed by "left".

Since the existence of ideals in an algebra & has been shown to be equiva-

lent to the existence of singular elements in 2, the following theorem is immedi-

ate.

THEOREM 7. Q, is a division algebra if and only if 12 contains no singular

elements.

4. Homomorphisms of G. The following result is an immediate consequence

of Theorem 4 and its analogue for left ideals.

THEOREM 8. If Q is congruent to a matrix of the form

iτ °l\ o o | ,

where T is a nonsingular m by m matrix, then the right paras trophic ideal B is

also a left paras trophic ideal. Conversely, if B is a right paras trophic ideal

determined by Q9 then if B is also a two-sided ideal, Q satisfies the above

condition.

Such an ideal will be called a parastrophic ideal, and Q will be said to have

P-rank m. While P-rank is not defined for every matrix, it is a property of every

symmetric matrix. Thus, if the characteristic of 9 is greater than n, the radical

of β is a parastrophic ideal. (It will be apparent shortly that this is true re-

gardless of the field characteristic since a semisimple algebra is a Frobenius

algebra.)

It does not follow that a matrix of 2 of P-rank m determines a homomorphism

HOMOMORPHISMS OF AN ALGEBRA 507

of G onto a Frobenius algebra of order m, for any commutative nilpotent non-

zero algebra contains proper parastrophic ideals. The following indicates a

necessary criterion.

LEMMA 1. If π is a homomorphism of U onto C, an algebra with an identity

element 1, then U contains an idempotent element e such that πe = 1. Further-

more, the set of left annihilators of e is contained in the kernel of the homo-

morphism.

This follows simply from the structure theory for algebras.

Now suppose the last n — m basis elements of U form a parastrophic ideal

13, and suppose that d has an idempotent element u such that du u 6 = 0,.

Then 13 will be called a regular parastrophic ideal,

THEOREM 9. A homomorphism of d onto a Frobenius algebra of order m has

as its kernel a regular parastrophic ideal of order n — m, and conversely if ID

is a regular parastrophic ideal of Cl of order n — m, then U — ID is a Frobenius

algebra of order m.

Suppose 0/ is a Frobenius image of U, with basis βι, , e m and kernel

θ spanned by em + ι, . . , en.

Then U possesses a nonsingular m by m parastrophic matrix Q$

and

is an element of 12 with Pπrank m. Hence 13 is a parastrophic ideal. By Lemma

1, IS is a regular parastrophic ideal.

The converse follows from the regularity of 6 and Theorem 6.

Thus, if Q is a parastrophic matrix of rank m9 if Q can be reduced to a

corner matrix by a change of basis of U, and if Q is associated with a linear

combination of the first m of the new basis elements, then Q determines a

homomorphism of U onto a Frobenius algebra. Furthermore, each homomorphism

of 0/ onto a Frobenius algebra may be determined in this fashion.

508 W. E. DESKINS

5. Radical-like ideals. A function f of & into the s e t of all ideals of & i s

called a radical function of Cl if the contraction of f to the difference algebra

C = U —/"(Q/) maps C onto the zero ideal. The ideal f(Q) is called a radical-

like ideal of CL

Let l be the set of all regular parastrophic ideals of U and let ΓL be an

element of Γ of minimal order, with the agreement that Γl is the zero ideal if

U is a Frobenius algebra and Q, if & is nilpotent. Then define / ( &) = ϊl.

T H E O R E M 10. f(&) = U is a nilpotent ideal of CL

If Cί is nilpotent the theorem is trivially true, so assume that & has the

radical K ^ Q,. Suppose K is of order r and that U is of order m. Let

Π n K = C .

Case 1. C = (0) . Let fl have a basis such that the first m basis elements

span \l while the last r span JC. By the definition of H there is an element

Q' of 12 of rank rc — m with its first m rows and columns composed of only zeros.

Now ϊl is isomorphic to a semisimple subalgebra of U — Kt so there is an element

Q" of 12 of rank m with only zeros in its last n — m rows and columns. Then

Q ' + Q " is nonsingular.

Case 2. C ^ ( 0 ) . Then 0/- C is a Frobenius algebra by the above work.

In either case

so that ϊi is contained in & and so is nilpotent.

One important property which Γl may lack is uniqueness. The question of

whether Yί is unique up to an & -isomorphism will now be considered and parti-

ally answered.

The following result indicates a significance of the U-isomorphism of two

minimal elements of ί .

THEOREM 11. Let ϊl and lΐi be minimal elements of r. Then a necessary

condition that U and tfL be &~isomorphic is that ϊi - C and lU - C be zero alge-

bras, where C = ft n ϊtl.

It may be assumed that C = (0) . Then let σ be an G-isomorphism from Ifl onto

ΐίl. If a and b are elements of ϊfl, then ϊi contains an element b ' such that


ab = a(σb') = σ(a δ ' ) = 0 .

The isomorphism of the minimal elements of ί for certain algebras will

stem from the following lemma.

LEMMA 2. If V and V are n by n matrices of rank m with elements from a

field o which contains at least w + 1 nohzero elements, then o contains an

element t ^ 0 such that U + tV is of rank at least m.

It will be sufficient to prove the result for m = n. Let D and E be nonsingu-

lar n by n matrices such that

DVE = / .

Consider the equation

det(D(U-xV)E)=άet{DUE-xI)**09

which is of degree n in the indeterminate %% Since o contains at least n + 1

nonzero elements, one of them does not satisfy this equation.

Let U and V be n by n matrices, and let

mean that the two matrices do not both have nonzero elements in the same row-

column position.

THEOREM 12. // two minimal elements ϊlχ and Π/2 of I are determined by

symmetric matrices Qι and Q2 of 2 of rank m% if o contains at least m + 1

nonzero elements, and if

<?lΛ<?2=0,

then ft — H ι is isomorphic with u - U 2 .

The cyclic modules

are of order my and the representations of U over these double U-modules yield

Frobenius algebras of order m which are images of U, isomorphic with U - H t

and & - H2 respectively. Let t be a nonzero element of o such that Q{ + tQ2 is

of rank m (since higher rank would contradict the minimality of the order of fli

510 W.E. DESKINS

and Π/2) Since Q^ and Q2 are symmetric

is a cyclic module of order mf and since Qγ

κQ2 ~ 0> (1) a n ( ^ (2) imply that

the mapping

is an (i-isomorphism between (^ * G, and ( ( ^ + ί(?2 ) * Ct. Similarly (? * Cί and

(ζ) t + tQ2 ) * fl are Ci-isomorphic. Hence Qχ * Q, and @2 * Ci are CUisomorphic

which implies that U — Hi and U — \l2 are isomorphic.

6. A remark concerning Frobenius algebras. While Frobenius algebras are

generally regarded as algebras with radicals of sufficiently small order, the

following indicates that their radicals must also be~ of sufficiently large order.

THEOREM 13. Let (λ be a Frobenius algebra bound to its radical K. Then

if & — K is of order m9 K is of order at least m. If K is a zero algebraf then &

is of order m.

By the results of Nakayama [9] the set of all elements of U which annihilate

K from the right is an ideal <C which also annihilates & from the left and has

order n - k = m9 where k is the order of K. Since (X is bound [4] to ίί,

£ cK,

hence m <_k9 and m = k if <C = Jί.

The consideration of bound algebras is, of course, sufficient since an alge-

bra may be written as a direct sum of a semisimple algebra and a bound algebra.

(This result is due to M. Hall [4] ) .

REFERENCES

1. R. Brauer, On hyperkomplex arthmetic and a theorem of Speiser, A SpeiserFestschrift, Zurich, 1945.

2. R. Brauer, and C. Nesbitt, On the regular representations of algebras, Proc.Nat. Acad. Sci., 23 (1937), 236-240.

3. G. Frobenius, Theorie der hyperkomplexen Groessen, S.-B. Preuss. Akad. Wiss.,

Berlin (1903), 504-537.

4. M. Hall, The position of the radical in an algebra, Trans. Amer. Math. S o c ,48 (1940), 391-404.


5. M. Ikeda, and T. Nakayama, Supplementary remarks on Frobeniusean algebras,Osaka Math. J., 2 (1950), 7- 12.

6. C.C, MacDuffee, Modules and ideals in a Frobenius algebra, Monatsh. Math.

Phys., 48 (1939), 292-313.

7. , On the independence of the first and second matrices of an algebra,

Bull. Amer. Math. Soc , 35 (1929), 344-349.

8. , The theory of Matrices, Chelsea, New York, 1946.

9. T. Nakayama, On Frobeniusean algebras, I, Ann. Math., 40 (1939), 611-633.

10. , On Frobeniusean algebras, II, Ann. Math., 42 (1941), 1-21.

11. , On Frobeniusean algebras, III, Jap. J. Math., 18 (1942),

12. , Orthogonality relations for Frobenius and quasi-Frobenius algebras,Proc. Amer. Math. Soc, 3 (1952), 183-195.

13. C. Nesbitt, On the regular representations of algebras, Ann. Math, 39 (1938),634-658.

14. M. Osima, Some studies on Frobenius algebras, Jap. J. Math., 21 (1951), 179-190.

15. G. Simura, On a certain ideal of the center of a Frobeniusean algebra, Sci. PapersColl. Gen. Ed. Univ. Tokyo, 2 (1952), 117-124.

THE OHIO STATE UNIVERSITY

THE MEASURE RING FOR A CUBE OF ARBITRARY DIMENSION

J.M.G. F E L L

1. Introduction. From Maharam's theorem [2] on the structure of measure

algebras it is very easy to obtain a unique characterization, in terms of cardinal

numbers, of an arbitrary measure ring. An example of a measure ring is the ring

of lίellinger types of (finite) measures on an additive class of sets. In this

note, the cardinal numbers are computed which characterize the ring of Hellinger

types of measures on the Baire subsets of a cube of arbitrary dimension.

2. Definitions. A lattice R is a Boolean σ-rίng if ( a ) the family Rx of all

subelements of any given element x form a Boolean algebra, and (b) any count-

able family of elements has a least upper bound. If, in addition, for each x in

R there exists some countably additive finite-valued real function on Rx which

is 0 only for the zero-element of R, then R is a measure ring. x A Boolean σ-

ring with a largest element is a Boolean σ-algebra. A measure ring with a

largest element is a measure algebra. The measure algebra of a finite measure

μ is the Boolean σ-algebra of μ measurable sets modulo μ null sets.

A subset S of a Boolean σ-algebra R is a σ-basis if the smallest σ-sub~

algebra of R containing S is R itself. R is homogeneous of order Cί if, for every

nonzero element x of R, the smallest cardinal number of a σ-basis of Rx is Cί,

We observe that R cannot be homogeneous of finite nonzero order. If it is homo*

geneous of order 0, it is the two-element Boolean algebra.

Let Cί be an infinite cardinal, / the unit interval [0, 1], and Ia the topological

product of / with itself Cί times (the Cί-dimensional cube). L will denote the

product Lebesgue measure on the Baire subsets of 7α, and M the measure

algebra of Z/ α . Then it is not hard to see that M is homogeneous of order

Cί. Maharam has in fact shown that it is, essentially, the only measure algebra

of order Cί.

1Our use of the terms 'measure ring', 'measure algebra', unlike Maharam's, refersonly to the algebraic structure. In this sense two measure rings are isomorphic if theyare isomorphic as σ-rings.

Received March 3, 1954.Pacific J. Math. 5 (1955), 513-517

513

514 J.M.G. FELL

3. Maharam's theorem and the characterization of measure rings.

THEOREM. (Maharam [2]) . Every measure algebra homogeneous of order

OC is isomorphic to M . Every measure algebra is a direct product of countably

many homogeneous measure algebras.

Consider now a measure ring R. A nonzero element x of R is homogeneous of

order Cί if the principal ideal Rχ is so. The cardinal function p of R will as-

sociate with each cardinal Cί the smallest cardinal number pa of a maximal

disjoint family of elements of R which are homogeneous of order α.

THEOREM, The measure ring R is determined to within isomorphism by its

cardinal function p.

Proof. If, for each r in an index set Kt Nr is a measure algebra, we may de-

fine a σ-ring

rβK

consisting of all functions φ on Kf such that ( a ) φ 6 NΓ for r G K9 (b) φ = 0

for all but a countable number of indices r. Union, intersection and difference

are defined in the obvious manner. N is actually a measure ring. Indeed, suppose

that φ E /V; and that the countable set of indices r for which φr ^ 0 is arranged

in a sequence r 1 ? r 2, . For each i, let m[ be a countably additive nonnegative

finite-valued function on NΓ. having value 0 only for the zero-element of Nr.

Then the function m, defined for subelements ψ of φ by

is countably additive, nonnegative, and finite-valued and has value 0 only for

Now for each cardinal α, let {%ag} (β < pa ) be a maximal disjoint family

of elements of R homogeneous of order Cί; let Raβ be the algebra of subelements

of %aβ By the definition of homogeneity, the %a n are all disjoint, and it is

almost trivial to show that

« s ΓUα/3a,β

THE MEASURE RING FOR A CUBE OF ARBITRARY DIMENSION 515

But, by Maharam's theorem, R o 2i M . Hence

R z ΠtΛί ( α )]

which depends only on the cardinal function p.

4. The measure ring of Hellinger types for the cube. Let F be an additive

class of sets. A finite measure μ on F is absolutely continuous with respect

to a finite measure v on F if all μ null sets are v null sets. If μ and v have the

same null sets, they are of the same Hellinger type. The relation of absolute

continuity furnishes an ordering of the Hellinger types under which, as is well

known, the latter form a measure ring. In this ring, the ideal of all Hellinger

types contained in the type of a measure μ is isomorphic with the measure

algebra of μ

We shall denote by R the measure ring of Hellinger types associated

with the additive class Bâ' of Baire subsets of the cube 7α; and shall obtain

the cardinal function characterizing R .

/ it \

LEMMA 1. R^^ R ° for any finite nonzero a.

Proof. This follows from Kuratowski's result [1] that any two complete

metric spaces X and Y having the same cardinal number are connected by a

one-to-one Borel mapping of X onto Y whose inverse is also Borel.

In view of Lemma 1, we restrict ourselves to cardinals OC which are in-

finite.

LEMMA 2. The total number of measures on B a is equal to or less than

ca

9 where c is the power of the continuum.

Proof. Ia is the set of all functions φ on a set H, of cardinal number OC,

to [0,1] . Let P denote a basis for the topology of Ia consisting of all subsets

of la of the form

E(φ£la

9 φβejβ for all β in G)

Φ

where G is a finite subset of H, and, for each β, Jβ is an open interval in /

with rational endpoints. Since CC is infinite, it is easily seen that the cardinal

number of P is OC.

516 J.M. G. FELL

Now suppose μ and v are two measures on β ' α ' which coincide on P. I claim

they must then be equal. If this is so, the lemma is proved, since the number

of real functions on P is c α .

Since P is closed under finite intersection, it is easily seen that μ and v

must coincide on finite unions of sets in P. Now let A be a closed Baire set

in Ia; A must then be a countable intersection of open sets. But, between A and

any open set containing it, we may by the compactness of /α, place a finite

union of sets in P. Passing to the limit, we find that μ and v have the same

value for A, Now by the regularity of μ and v, their coincidence is assured for

all Baire sets.

LEMMA 3. For each cardinal γ which is either 0 or infinite and for which

γ <_ α, there are at least ca disjoint measures on Z? whose measure algebras

are homogeneous of order γ.

Proof. With each point φ in Ia associate the measure μ , of a unit point

mass at φ. Any two such points φ and φ can be separated by Baire sets, hence

the Hellinger types of μ, and μ, are disjoint. Since there are c α points in 7α,

the lemma is true for γ = 0.

For any infinite cardinal y, divide //, the index set of Ia (see proof of Lemma

2), into two disjoint parts M and /V, of cardinal number γ and Cί respectively.

Let / and K be the γ- and CX-dimensional cubes with index sets M and N res-

pectively. Since M u N = H$ two points φ and φ, in / and K respectively, define

a point of Ia which we call φ u φ, for which ( φ u φ)r = φr or φr according as

r G M or r E N. We fix φ in K; and for each Baire subset A of /α, let Tφ{A) be

the subset of / consisting of those φ for which φ u φ £ A.. I claim that Tψ is

a σ-homomorphism of Bâ' onto all Baire sets in /.

That Tψ is a σ-homomorphism is evident. Now Bâ* is the smallest additive

class of subsets of Ia which contains P (defined in the proof of Lemma 2).

That every Baire set A in Ia maps into a Baire set in / will follow if sets in

P go into Baire sets; but the latter is evident. A corresponding argument in /

shows that all Baire sets in / are maps of Baire sets in /α. Hence the claim

made for Tψ is correct.

Now let I / 7 ' denote the product Lebesgue measure defined on the Baire

sets of /. For each φ in Kf and each Baire set A in /α, put

Evidently the measure algebras of μ, and Lry' are isomorphic; the former is

THE MEASURE RING FOR A CUBE OF ARBITRARY DIMENSION 517

t h e r e f o r e h o m o g e n e o u s of order y. F u r t h e r , if φί9 ψ2 £ K9 r E /V, ψîr) j^ ψ2(r),

and A i s t h e B a i r e s e t of a l l φ in / α w i th φ(r) = ψιir), we s e e t h a t

It follows that μxpι and μψ2 are disjoint whenever ΦlfΦ2 ^ ^ a n <^ Φι ^ Φ2

The fact that there are ca elements in K completes the proof of the lemma.

THEOREM. If Cί is an infinite cardinal^ the cardinal function p characteriz-

ing the measure ring R of Bellinger types of measures on the Baire subsets

of the Cί-dimensional cube is given by:

ca if γ is a 0 or infinite cardinal with γ <_ Cί;

0 otherwise.

Proof. Since P, a σ-basis of B (see proof of Lemma 2), is of cardinal

number Cί, we must have p = 0 for y > Cί. If y is 0 or infinite, and y <_ Cί,

Lemmas 1 and 2 prove the existence of a maximal disjoint family of elements

of R homogeneous of order y, of cardinal number exactly ca. That ca is the

smallest possible cardinal number of such a family, follows easily from the fact

that no element of R intersects more than a countable number of disjoint

elements of R{a\

R E F E R E N C E S

1. C. Kuratowski, Sur une generalisation de la notion d'homeomorphie, Fund. Math.,22 (1934), 206-220.

2. D. Maharam, On homogeneous measure algebras, Proc. Nat. Acad. Sci., 28 (1942),108-111.


THE NORM FUNCTION OF AN ALGEBRAIC FIELD EXTENSION, II

H A R L E Y F L A N D E R S

1. Introduction. In our previous paper [3], we consider the general norm

of a finite extension K of an algebraic field k. We proved that this form is the

(n/m)th power of an irreducible polynomial in λ[λΠ, where m is the maximum

of the degrees of the simple subfields k(θ) of K over k. The proof of this result

used a considerable amount of the heavy machinery of the theory of algebraic

extensions: the maximal separable subfield, conjugates, transitivity of the norm,

etc. Using only the fact that the general norm is a power of an irreducible, we

obtained a characterization of the norm function NK/k ι n terms of inner proper-

ties.

In the present paper we shall approach these matters from a different point

of view. We shall give an entirely different proof that the general norm is a

prime power—this one based on very little field theory and completely rational.

From this, as noted above, the intrinsic characterization of the norm function

follows. We shall then use this to derive certain theorems in field theory, such

as the transitivity of the norm.

Section 2 contains some preliminary results on polynomials and their norms

and the details of proof for certain results used in [3] , In § 3 we prove the main

result and in § 4 we give some applications.

2. Tool theorems. We shall be dealing with polynomial rings &[X]in inde-

terminates X = (Xί9 ,Xr) and shall take for granted the fundamental fact that

such rings are unique factorization domains [ l , p . 39]. The following is well

known, but we include it—as we do several of the results of this section—for

completeness.

LEMMA 1. Let f (X), g(X) £ k[X] and suppose f and g are relatively prime.

Let k <^K so that k[X] < K[X], Then f and g are still relatively prime when

considered as elements of the extended ring K[X].

Received October 19, 1953.

Pacific J. Math. 5 (1955), 519-528

519

520 HARLEY FLANDERS

For the case r = 1 of one variable, this is so because of the Euclidean

greatest common divisor algorithm.

In general, we suppose H{X) is a common factor of f and gs H(X) EK[X\

Without loss of generality, we may assume that H has positive degree in Xr,

We form the fields of rational functions,

K == K (x i, , xr_ i), k = k (x i, , xτ. i),

and the p o l y n o m i a l s

/ ( T) = / ( % i , •• , x r - i > T), g~( 71) = g ( % i , •• , * Γ . i , 71)

of t h e r ing k[T]. T h e s e p o l y n o m i a l s h a v e a non-tr iv ia l factor H(x9 T) in K[T],

h e n c e by the c a s e r = 1, they h a v e a n o n - c o n s t a n t factor hι(x$ T) £ k[T]:

J(T) = hί(x,T)fι(T), ^(T) = hί(χ,T)gι(T).

Here hi, fy, gjβre polynomials with coefficients rational functions over k in

x = {xι, ' , xr.i ). Multiplying by a suitable denominator q (x ), we obtain

q(x)f(x, T)=h(x, T)f2(x, T), q(x)g(x, T) = h(x, T)g2(χ, T),

where all terms are polynomials. This implies

q(Xi,...,Xr.i)f{X)-h{X)f2(X), q(Xι,' ',Xr.ι)g(X)"h{X)g2(X).

Since h(X) actually involves Xr, it follows from unique factorization that some

irreducible factor of h must divide both f and g,

LEMMA 2. Let k be a field, JO an integral domain such that k < D, and such

that if JO is considered as a linear space over k$ then £ is finite dimensional.

Then c is a field.

Proof. Cf. [2, p. 75 ]. If α £ JO and a 0, then the mapping b—> ab is a

one-one linear transformation on JO into JO. Since JO is finite dimensional and

rank plus nullity equals dimension, it must map JO onto £>. Thus 1 = ab for some

b then a has an inverse.

LEMMA 3. Let [K:k] = n and ωί9 ,ωn be a basis of K over k. Then

[K(X):k(X)] = nand(ω) is a basis of K(X) over k(X).

Proof. Let

THE NORM FUNCTION OF AN ALGEBRAIC FIELD EXTENSION, II 5 2 1

o = k(X)ωt + . . . +k{X)ωn.

Then o is a finite dimensional integral domain over k(X) and

k(X) <a

By Lemma 2, JO is a field; since

K = kωt + + kωn < o ,

we have

K(X) = K-k(X) < a ,

hence JO = K(X). It follows that (ω) spans K(X) over k(X). But it is clear

(by equating coefficients) that (ω) is linearly independent over the rational

function field k(X).

We introduce the norm in this way. If [K:k] = n and A G K9 then Nj^/Â is

the determinant of the linear transformation B —* AB on K over k Specifically,

if cύij * , ωn is any basis of K over /c, and

then

We similarly define the irace

Sκ/k A = Σ α ί t

for later purposes. The rules

= (NK/kA){NK/kB)9 Sκ/k(A+B)=Sκ/kA+Sκ/kB,

= αn. Sκ/k(a) =n - a,

follow immediately.

We form the fields K(X), k(X) so that also [K(X): ft (Z) ] = n and we

may discuss

522 HARLEY FLANDERS

for R(X) G K(X). We shall use the abbreviation

as we did in [3] since this can hardly lead to confusion.

LEMMA 4. Let

F(X)eK[X] and f (X) = Nκ/kF(X).

Then

f(X)£k[X]

and F (X) divides f (X) in the ring K [ X ].

Proof. We write

ΣA(a) A'(α)

where A G K and X(a) is a monomial in X — (Xγ, , XΓ). We have

hence

I = Zô A!(α) ω, = έ^f..\X)cύj,l>] J If J

f(X)~Nκ/kF=\f..\ek[X]t

where f . . G ί ; [ ^ ] . Thus

which settles the first point. We may also write

which implies

On expanding the determinant we soon see that F(X) does indeed divide f(X).

LEMMA 5. //


F(X),G(X)eK[X], h(X)ek[X],

and

F(X) = G(X) (moάh(X)),

then

Nκ/kF(X)^Nκ/kG(X) (modh(X)).

Proof. We may write

F(X) = G(X) + h(X)Q(X)

with Q(X) E K[X\ As above, we have

ωj, G(X)ωi=Σgij(X)ωj,

i t h fij 8ij> Ίij G * [ ^ l T h u s

and therefore

7V(F) == 1/ .| = \gij + hq.j\ = l g ι 7 l = ^ V ( G ) U o d A U ) ) .

LEMMA 6. Let F(X) be an irreducible polynomial in K[X]. Let f (X) =

/Vχ/^F(Z) and suppose that g{X) is any non-constant divisor of f(X) in

k[Xl Then F{X) divides g(X).

The case r = 1 is given in [4, p. 19].

Proof. If r = 1 and F(X) does not divide g(X), then we can find polyno-

mials U(X), V(X) eK[X] such that

Thus

U(X)F(X) = l (moάg(X)).

By Lemma 5 we obtain

524 HARLEY FLANDERS

u(X)f{X) = l ( m o d g U ) )

which is clearly impossible.

In the general case we may suppose that the degree of F in Xr is positive

and pass to the rational function fields k — k(x)y K = K(x)y where x = (x ι ? ,

xΓmi). The usual unique factorization argument shows that F( T) = F(x i9 •• ,χr-ι* T) is irreducible in K[T]. For the norm we have

The polynomial ~g{T) = g{xι, ,xrm\$T) divides f (T) in k[T]; it follows

from the case r — 1 that F(T) divides g(T):

We multiply by the denominator of // to arrive at a relation of the form

Since F(X) is irreducible, this implies that F(X) divides g(X),

THEOREM 1. Let F(X) be irreducible in K[XI Then {(X) = Nκ/k F{X)

is a power of an irreducible polynomial in k\_X\

Proof. If p(X) and q(X) are irreducible factors όί f(X) in k[X]9 then by

Lemma 6, F(X) divides both p{X) and q(X). This implies, by Lemma 1, that

p(X) -q{X). Hence f {X) has only one distinct irreducible factor.

NOTE 1. In the proofs of both Lemma 1 and Lemma 6, the reduction of the

case of general r to the case r = 1 could have been effected by the Kronecker

device of substituting suitable powers of a new variable T for the X(9 since in

these statements we dealt with only a finite number of fixed polynomials and

their divisors, all of bounded degree.

N O T E 2. Lemma 1, for the case in which [K:k] ~n9 is an immediate con-

sequence of Lemma 4. For if H{X) E K{X) and H{X) is a non-constant common

divisor of f and g, then we have f=HFί, g = HGί9 and thus

fn = N(f) = N(H)N(Fι\ gn = N{H)N{Gι).

But H divides Nκ/kH, hence N{H) is non-constant. This is clearly impossible

when f and g are relatively prime.


Once Lemma 1 is proved for finite extensions, it can be proved for arbitrary

extensions by the use of a transcendence basis.

3. The general norm. Let [K:k]-n and let ω 1 ? , ω n be a basis of K

over k. As in [3] , we form the general element

Ξ= o)ιXι + + ωn Xn eK[X]

and the general norm

NN/k(~)£k[X]

which is a form of degree n

THEOREM 2. The general norm is a power of an irreducible polynomial in

k[X].

Proof. The general element E is a linear form in K[X], hence irreducible;

Theorem 1 now applies.

From this now follow the results of § 3 of [3]; we state the following in-

stance.

THEOREM 3. Let [K:k] -n and let φ be a function on K into k with the

following properties:

(1) φ(AB) = φ(A)φ(B).

(2) φ(a)=an.

( 3 ) φ ( Σ αj CU( ) = / (a i, , an),

where f is a polynomial of degree at most n. Then φ(A) -Nj^/kA for all A in

K.

4. Applications. Let k <.L < K, where K is a finite extension of k, and

consider the function

A—>NL/k[Nκ/LA]

on K into k. Evidently this satisfies the properties (1,2,3) of the theorem

above, so we obtain

526 HARLEY FLANDERS

Next, let [K: k] = n and let A G X. The /"je/rf polynomial of 4 is

It is clear that / ^ ( Ό = 0 and that fA(T) is the minimum polynomial of A in

case X = k{A )—since 1, /4, , /471"1 is a basis in that case. If K >_ L > k9

then

fA,κ/k{TUNκ/k(T-A)=NL/k[Nκ/L(T-A)]

Especially HA GL, then

Here is another consequence; if K > L >_k and A £ K, we have

For if [K:fc] = r, then

Our statement follows at once from this and the following lemma.

L E MM A 7 . L e t [ K : k ] = n and

f ( T ) = Γ + A x Tr'1 + + A r e K I T ] .

T h e n

This is proved by slightly modifying the proof of Lemma 4.

Finally we derive the familiar expressions for the norm and trace in terms

of conjugates. Let [K: k] = n and let K < ί/. Suppose σ1 ? , σw are n not

necessarily distinct isomorphisms over k on K into ί/ with the property that

whenever h (A^, , Xn) is a symmetric polynomial in k [A] then λ ( OΊ (/4 ), ,

σn(A )) E & for all A £ K. We consider the mapping

THE NORM FUNCTION OF AN ALGEBRAIC FIELD EXTENSION, II 527

on K into k. This sat is f ies properties ( 1 ) and ( 2 ) of the las t theorem. To show

that it also sat is f ies the third property, we let ω t , , ωn be a bas is of K over

k and let A - ΣL α; ω^ be an element of K, α t £ k. Then

ί=l J

where f is a form of degree n in aγ9 9an whose coefficients are, until we

say more, in U If k is infinite, one finds that these coefficients are in k from

the fact that f (aχ9 9an) G k for all vectors ( α l 5 , an); when & is finite,

then K — k(B) is simple over k9 and we may use 1, Bf •• 9Bn~ for a bas i s .

Then the coefficients of f are symmetric in σx ( B ) , , σn( B ), and hence are

in &. At any rate we obtain

If F(Γ) = Σ/4 t Γ, we set

and make the obvious extension to rational functions. A similar argument to

that above implies that

h(Rσι(T),.->,Rσn(T))ek(T)

when h (X) is symmetric in X = (XΛ , , Xn ), h (X) £ k[X], and R(T) E K(T).

It follows that the formula for the norm as a product (of conjugates) is also

valid in K(T) over k(T), hence in particular

and by comparing the second coefficients,

REFERENCES

1. A. A. Albert, Modern higher algebra, Chicago (1937).

2. N. Bourbaki, Algebre, Chapitre V, Corps commutatifs, Paris (1950).

3. H. Flanders, Norm function of an algebraic field extension, Pacific J. Math.3 (1953), 103-113.

528 HARLEY FLANDERS

4. H. Weyl, Algebraic theory of numbers, Princeton (1940).

THE UNIVERSITY OF CALIFORNIA, BERKELEY

ON THE CHANGE OF INDEX FOR SUMMABLE SERIES

D I E T E R GAIER

1. Introduction. Assume we have given a series

(1.1) α 0 + a l + a2 + + an +

and consider

(1.2) b 0 + b i + b 2 + + b n + w i t h b 0 = 0 a n d b n = a n . ι ( n > _ l ) ;

denote the partial sums by sn and tn, respectively. Since sn = ί Λ + i , the con-

vergence of (1.1) is equivalent to that of (1.2). However, if a method of sum-

mability V is applied to both series, the statements

(1.3) (a) V-Σan=s (b) V-Σbn=s1

need not be equivalent (for example, if F i s the Borel method; see [4, p. 183]).

If V(x;sv) and Vix tp) denote the F-transforms of the sequences { sn \ and

{tn \, respectively, it is therefore interesting to investigate, for which methods

V and under what restrictions on { an \ the relations

(1.4) ( a ) V(x;sv)^ K . x* ( b ) V(x; tv) ~ K . χ<?

(x—> XQ , K c o n s t a n t ; q >_ 0, f i x e d ) 2

are equ iva lent .

The c a s e s V^C^ ( C e s a r o ) and V~A ( A b e l ) are quickly d i s p o s e d of

( § 2 ) , while V~E ( g e n e r a l E u l e r t ransform) and V~B ( B o r e l ) p r e s e n t some

i n t e r e s t ( § § 3 - 5 ) .

2. THEOREM 1. The statements (1.4.a) and (1.4.b) are equivalent for

XWe shall always let L ° ° s o n = Σ, an.

x—y XQ through values depending on the method V,

Received December 1, 1953. This work has been sponsored, in part, by the Officeof Naval Research under contract N5ori-07634.

Pacific J. Math. 5 (1955), 529-539529

530 DIETER GAIER

V = Ck(k > - 1 ) and V ~ A. 3

Proof. If

s < f c ) - < * ( * . „ ) . ( " +

and

we have by definition of the Cesaro means

(2.1) (l-x)k+ι Σ.τlk)xn

the series being convergent for | x \ < 1. The proof of Theorem 1 now follows

from the inner equality in (2.1) and the relation

γ(k) o(k) S^^n τ ι - 1 " - 1 . .

\n —> oo .ίn + k\ jn + k\ /n-l + k\

\ n I \ n I \ n - 1 /

3. Let g(w) = Σ,γnwn be regular and schlicht in | w \ £ 1, and assume

g ( 0 ) = 0 , g ( l ) = l. Then the ^-transforms of Σ,an and 22b n are obtained

by the formal relations [ 5 ]

Σ,anzn = Σ,an[g(w)]n = Σ,0Lnw

n; E(n;sp)= ^ av

(3.1) (τι = 0 , l , . . . ) .

Σbnzn=Σbn[g(w)]n=Σβnw"; E(n;tv)=Σ β*

THEOREM 2. The statements (1.4.a) and (1.4.b) are equivalent for V - E.

Proof. First we note that if either

E{n;sv)^0(n^) or E U; tv) = O(n^) (n—*ω),

}For q = Osee [4, p. 102].

ON THE CHANGE OF INDEX FOR SUMMABLE SERIES 5 3 1

then the formal relations (3.1) are actually valid for \w \ < 1 and also

( 3 . 2 ) Σ,βnwn= Σbn[g(w)]n=g{w). Σan[g(w)]n = g(w). Σ Clnw

n

(\w\ < 1) .

Denote by Anf Bn9 Cn the partial sums of Σdn, Σβn, Σγn, respectively.

We assume first

E (n; sv) - An ~ K n^ (n —> oo) .

Then, since by (3.2) Σ,βn is the Cauchy product of Σ,(Xn and Σ y π , we have

E(n;tv) = Bn=γnA0 +γnmlAί + ... + γιAn_ι

and for ^ >. 1

( 3 . 3 ) — = — A 0 + γ _ . _ + ..,. + γ(u-l)q

For the matrix cnV in this transformation of the convergent sequence \Ann"^

we have clearly

lim cnV = 0 ( v = 0, 1, •)«

n —• oo

Furthermore

n-1 vq \γn\ n oo

finally we prove

lim

For ςr = 0 this follows from

532 DIETER GAIER

for q > 0

nH

£ ί/n v\? fn-v-l\1

and the last term is a positive regular transformation of the sequence { Cn\

tending to g ( l ) = 1, whence

^2. cnv —> 1 ( n —>oo) .

v

Therefore the transformation ( 3 . 3 ) of \ An n"^ \ converges to K, which proves

Bn - K-n? ( Λ — > ( » ) .

Assume on the other hand Bn ^ Knq in —» oo). Putting w = 0 in ( 3 . 2 ) , one

obtains βQ = 0, so that

Zanwn = [g{w)}-1 Σβnw

n=w[g(w)Yι

is regular in \w\ < 1. Furthermore the expansion of the function w[g{w)]~ι

for w = 1 converges absolutely to 1, s ince w = 0 is the only zero of g{w) in

Iu; I £ 1. An argument similar to the one above shows then that Bn^.ί gί Knq

(n—> QQ) implies An ^ Kn^ (n —>oo), which completes the proof of Theorem

2.

We add a few remarks about the assumptions on the function z = g(w) by

which the E-method is defined.

a. Theorem 2 becomes false if only regularity of g{w) in \w\ < 1, and con-

tinuity and schl ichtness in \w\ <_1 are assumed. For there exist such functions

g(w) whose power ser ies do not converge absolutely on | w \ = 1 (cf. [ 2 ] ) .

Therefore in ( 3 . 2 ) one could find a convergent Σ α n whose transform Σ,βn

diverges.

b. All that was used about the function g(w) in the proof of Theorem 2

was that the power ser ies of g(w) and of w[g{w)Yι converge absolutely to

the value 1 for w = 1. This can be guaranteed by the weaker assumption that

g(w) with g ( l ) = l and g ( 0 ) = 0 is regular in \w \ < 1, continuous and schlicht

ON THE CHANGE OF INDEX FOR SUMMABLE SERIES 533

in \w\ <. 1, and that the image of \ w \ = 1 under the mapping g(w) is a recti-

fiable Jordan curve. Because then

< ooΪ27T \g\e^Jo

and hence ϋL \γ \ < oo [8, p. 158]; on the other hand also

/ \G'{eιφ)\dφ < oo,Jo

where

I' giw) -wg'(w)£'(„,)_[-£_] =ίg(w)]2

so that also the power series of G(w) converges absolutely to the value 1 for

c. If

g(w) ~ w[(p + 1 ) - pw]~ι (P >_ 0> fixed )

one has E - Ep as the familiar Euler method of order p, for which Theorem 2 is

known in the case q — 0 [4, p. 180],

d. The function

g(w) = (2 - « , ) - 2(1 -wΫΛ ( g ( 0 ) = 0 )

leads to the method of Mersman [6] , as Scott and Wall showed [7, p. 270 ].

Here Theorem 2 is also applicable, since the more general conditions about

g{w) in remark (b) are satisfied, as is readily seen.

4. The Borel method is defined by the transformation

svxv

-χΣ ( x > o ) ,e

where the power series is assumed to define an entire function. It is known

that B(x sp)—>K [x—»oo) implies B{x;tv) —>K (x —»co), but not con-

versely [4, p. 183]. We now prove more generally

534 DIETER GAIER

THEOREM 3. The relation

B(x;sv) ~ Kx^ {x—> oo)

implies

B{x\tv) Ξί Kx^ (x—>oo).

Proof. We have for x > 0 [4, p. 196]

v\ {v+ 1)1

B (t; sv )<; f v /*r , * B(t:sv)-^- dt-χ-1 I e-^-'h* —dt.v\ Jo t^

This transformation of the convergent function B{t;sv)t"^ (t—»oo) by means

of the 'matrix

, (0<t<x)

is regular, since

I \c(x9t)\dt—*0 (%—> oo t u t 2 > 0, fixed)Jtγ

and

Γx foe / t \ q

\c(x,t)\dt= I c(x, t)dt =e"x e Ί - l dt—>1 (x—>oo).Jo Jo \x I

Therefore B(x;tv) ^ Kx^ (x—> oo).

We discuss now the converse of Theorem 3.

THEOREM 4. The relation

B(x;tv) z Kx* U—» oo)

implies

ON THE CHANGE OF INDEX FOR SUMMABLE SERIES 5 3 5

B(x;sv) 21 Kx^ (x—>oo),

if

( 4 . 2 ) l im s u p \an \ί/n < oo,

that is, if the series 2Lanzn has a positive radius of convergence.

Proof. Using (4 .1) we have for % > 0

Fix) ^x'^B(x;tv)=xê"x [% etB (t; sv)dt.

J o

C o n s i d e r n o w F ( x ) a s f u n c t i o n o f t h e c o m p l e x v a r i a b l e x f o r K { x ) >^ 1 . T h e n

( 4 . 2 ) i m p l i e s \tn\ <_ Mn f o r s o m e c o n s t a n t M > 0 a n d h e n c e i n H ( x ) >_ 1

and also

(4.3) \F(x)\ < aeP\x\ H(x)

for positive constants α and β. Hence one knows that

F(x)—>K (x—>+

implies

F'(x)—>0 (x—>+ω

t h a t i s ,

[XB(t;sv)dt ί - 1 - !

from which the result follows.

5. We now show that Theorem 4 is best possible in a certain sense.

4 I f F{χ) i s r e g u l a r in K(x) > 1 a n d ( 4 . 3 ) h o l d s , t h e n Fix)—>A ix — > + o o ) im-p l i e s F'(x)—>0 ix—>-f CXJ). T h i s lemma w a s u s e d a l s o in [ 3 ] , w h e r e T h e o r e m 4 w a sp r o v e d for q = 0.

536 DIETER GAIER

THEOREM 5. In Theorem 4 the Condition (4.2) cannot be replaced by

(5.1) l i m s u p ne \an\Wn < oo ( e > 0 ) .

For the proof we need the following

L E M M A . For every β > 1 , there exists a n entire f u n c t i o n f ( z ) of order

β satisfying

(5.2) / U ) _ > 0 U ^ + α>)f/'(*)-/-> 0 U — > + α>) U = *

Proof. Put α = /3"ι and consider the Mittag-Leffler function

which is an entire function of order Cί" = β. Let m be the integer with

α α< m < + 1.

1 - α 1 - α

We first study the derivatives of Ea( z) of order 1, 2, , m on the line arg z -

OLπ/2 for large | z |. For these z (assume for definiteness | z | > 2) one has

[ 1 , pp. 272-275]

(5.3) £ . < * > - — / V ' ' - 2 - + i β ' l Λ \2πiCl JL t - z α

the path L being

£ = re I oo > r > 1, Cίπ > φn > — I, t = e ( — ώn < φ < + φn ) ,i — * ' o 9/ ^ — — r u 7

t = reiΦo (1 < r < oo);

ί ι / α is the branch which is positive for t > 0. The A th derivative of the integral

part in (5.3) can then be estimated as follows

Λ/a k\* —dt

k\ Γ , . " a , |Λ

2πa\z\k+i/

A/a\el I

| l - (


since for our values of z one has | 1 - (t/z) | >. δ > 0 and on the straight line

segments of L

— e c o s α with cos — < 0 .Cί

Therefore

-j

£ ' ( 2 ) = o ( D + — ez z(X 9

a2

ι/a-1

1a

(5.4)α3

£ > - ι > ( 2 ) = o ( l ) + —

Now we consider the function

which is again an entire function of order CC1. For \z \—» oo on arg z = dπ/ 2

we have by (5.4)

am

however

α m + 1

and herein | e ε l / α | = l and ( ( l / α ) - l ) r o - l > 0, so that F ' ( z ) - / * 0

( I z I—> co on arg z = a n / 2 ) . For the lemma it is therefore sufficient to take

538 DIETER GAIER

Proof of Theorem 5. Define the { an \ of (1.1) by

r< \ fx -t <Γ a v t V i ίx -ί / ^ 7/ ( % ) = / e

Σ L ώ = / e

ι a(t)dt,Jo v\ Jo

w i t h t h e fix) of t h e a b o v e l e m m a a n d /3 = ( l - € ) " 1 . S i n c e fix) i s of o r d e r

β > 1, s o i s o ( ί ) , a n d t h e r e f o r e [ 1, p . 2 3 8 ] 5

anλ / IΛ

suplim sup n ^ — e lim sup n" | α π | < oo ,

that is, (5.1) is fulfilled. Furthermore

/ ( % ) — > 0 (%-^ + oo),

which is equivalent to

B(x;tv)—>0 {x—> + oo).

However, in order that

B(x;sv) —>0 (x—> + oo),

it would be necessary and sufficient to have [4, pp. 182-183]

emχa(x)=f'(x)—*0 (x—>+oo),

which by our lemma is not fulfilled. So we have given an example of a ser ies

Σ o n for which B(x;tv) —» 0 (x —» +oo) does not imply B(x;sv) —> 0

(x —> + oo) and for which ( 5 . 1 ) holds.

Prof. Lδsch (Stuttgart) suggested to me the relation to the coefficient problemfor entire functions.

REFERENCES

1. L. Bieberbach, Lehrbuch der Funktionentheorie, 2. ed., vol. II, Leipzig, 1931.

2. D. Gaier, Schlichte Potenzreihen, die auf \ z \ = 1 gleichmassig, aber nicht absolutkonvergieren, Math. Zeit. 57 (1953), 349-350.

3. , Zur Frage der Indexverschiebung beim Boreί-Verfahren, Math. Zeit.58 (1953), 453-455.

4. G. H. Hardy, Divergent series, Oxford, 1949.


5. K. Knopp, Uber Polynomentioicklungen im Mittag-Leffίerschen Stern durch An-

wendung der Eulerschen Reihentransformation, Acta Math. 47 (1926), 313-335.

6. W. A. Mersman, A new summation method for divergent series, Bull. Amer. Math.Soc. 44 (1938), 667-673.

7. W. T. Scott and H.S. Wall, The transformation of series and sequences, Trans.

Amer. Math. Soc. 51 (1942), 255-279.

8. A. Zygmund, Trigonometrical series, Warsaw, 1935.

HARVARD UNIVERSITY

COMPLETE MAPPINGS OF FINITE GROUPS

MARSHALL HALL AND L. J. PAIGE

1. Introduction. A complete mapping of a group G is a biunique mapping

x —>®(x) of G upon G such that x ®(x) =* η(x) is a. biunique mapping of G

upon G. The finite, non-abelian groups of even order are the only groups for

which the question of existence or non-existence of complete mappings is un*

answered. In a previous paper [4] , some progress toward the solution of this

problem has been made. We shall show that a necessary condition for a finite

group of even order to have a complete mapping is that its Sylow 2-subgroup be

non-cyclic, and that this condition is also sufficient for solvable groups. We

shall also prove that all symmetric groups Sn(n > 3) and alternating groups

An possess complete mappings. In the light of these results the following con-

jecture is advanced:

CONJECTURE. A finite group G whose Sγlow 2-subgroup is non-cyclic

possesses α complete mapping.

It is interesting to compare this conjecture with the results of Bruck [2, p.

105].

2. Complete mappings for the symmetric and alternating groups. The follow-

ing theorem is a generalization of Theorem 4, [4] and will be necessary for

considerations of this and other sections.

THEOREM 1. Let G be a group, H a subgroup of finite index {G:H) =k .

Let u\, U29 , w/c be a s e ί °f elements of G that form both a right and left

system of representatives for the coset expansions of G by H. Let S and T be

permutations of the integers 1, 2, , k such that

Ui(us{i)H) =uτ{i)H, i = 1,2, . . . , & .

1ΓΓhe restriction that the index be finite is unnecessary. However, P. Bateman [ l ]has shown that all infinite groups possess complete mappings and so we have chosenthe present restriction for simplicity. In fact, the restriction that G be finite wouldseem appropriate.

Received December 18, 1953. The work of L. J. Paige was supported in part by theOffice of Naval Research.

Pacific J. Math. 5 (1955), 541-549541

542 MARSHALL HALL AND L. J. PAIGE

Then, if there exists a complete mapping for the subgroup H$ there exists a com-

plete mapping of G.

COROLLARY 1. Let G be a factorizable groups that is, G = A B9 where

A and B are subgroups of G with A n B = 1. If complete mappings exist for A

and B9 then there exists a complete mapping for G.

COROLLARY 2. If H is a normal subgroup of G, and both H and G/H pos-

sess complete mappings then G possesses a complete mapping.

Proof. By hypothesis,

(1) G = u\H + u2H + • + u^H = Hui + Hu2 + + Huk

and thus the equation

(2) i t s ( i ) P = p* i t [ s ( j ) t p ] , ( i = 1,2, • • • ! * ) , P€H,

uniquely defines p* and ^fs( ) π as functions of p and ι. Here, wr<j/.\ i = ut for

some 1 <_ t fC k. Moreover, p is uniquely defined by p * and i, for if

usϋ)Pι

then we would have

Since the w's form a system of representatives this would imply

u [ s ( i ) , p ι ] " U [ s ( ) .p a ]

and consequently pί = p 2

We have assumed that there exists a complete mapping for H; hence, there

is a biunique mapping ®ι of H upon H such that the mapping ηγ{p) = p Θ i ( p )

is a biunique mapping of H upon H.

Let us define a mapping of G upon G in the following manner:

O)

where p, p*, w[ s( ι ) 1 a r e defined by ( 2 ) .

COMPLETE MAPPINGS OF FINITE GROUPS 543

In order to show that Θ is biunique, assume that

ΘUίP*) = eu / P*).

Then,

u[sU),Pι] ' ei{Pi] = uίs(j),p2] ' &ι{P2

)I

and t h i s can h a p p e n o n l y w h e n u[sd) ]~u[s(~) ] i m p l y i n g Θ t ( p t ) = Θ L ( p 2 )

or p t = p 2 . Now,

and it would follow from (2) that i =/. If G is finite we may conclude immedi-

ately that Θ is a biunique mapping of G upon G. If G is infinite, we note from

(2) that if p is kept fixed, then as i ranges over 1, 2, . ., k; u[s(^ l ranges

over all coset representatives. Thus for any element ut p', we first find p

from p ' = Θ i ( p ) ; and then holding p fixed we vary i to find the p* such that

ιι . v . p = p* . Ufr For this i and p* we have

ΘUjp*) =Mt ®ι(p) =ut p\

and every element of G is an image of some element of G under the mapping Θ.

Let us now show that Θ is a complete mapping for G. Consider

η(uiP*) =uiP* Θ U p*) = ttip* " [ s (0 ,p] ' Θ ^ P ^ =uiuS(i) ' P Θ l ( p )

First, if ηiuip*) = ηiujp*), we have

(4) ^ ^ ( ί ) P i Θ ι ( P i ) = = w / z x S ( / ) P 2 Θ ι ( P 2 ) ' O Γ uT(i)H = uT(j)H>

and this is impossible unless i = /. Consequently from (4),

P 1 Θ 1 ( P l ) = P 2 θ 1 ( p 2 )

and Θ t being a complete mapping implies pγ - p2. Again the finite case is

completed and if G is infinite we note that there is but one i such that UiUς,ί.\H —

Uj>ί\H and the subsequent solution for p* is straightforward.

Corollary 1 follows from the observation that the elements of A form a sys-

tem of coset representatives satisfying the hypothesis of the theorem.


Corollary 2 is proved by noting that if

in G/H, then

We will use Theorem 1, to show that an earlier conjecture [4, p. 115] con-

cerning complete mappings for the symmetric groups Sn(n > 3) was wrong.

THEOREM 2. There exist complete mappings for the symmetric group Sn if

ifn > 3.

COROLLARY. (See conjecture [4, p. 115]). There exist Latin squares

orthogonal to the symmetric group Sn for all n > 3.

Proof. The proof will be by induction and we note first that S3 has no com-

plete mapping [3, p. 420], Thus we must exhibit a complete mapping for S 4 .

We may express S4 = A B9 where

A=\\9 (123), (132) ! ,

B = U , (12), (34), (12M34), (1324), (1423), (14)(23), (13)(24) },

are subgroups of S4 with AAB = 1. Moreover, there exist complete mappings for

A and B given by:

β ( l ) = l, 6(123) = (123), Θ(132) = (132)

for A; and

Θ ( l ) = l, Θ(12) = (34), Θ(34) = (1324), Θ( 12)(34) = (13)(24)

Θ(1324) = (14)(23), ©(1423) = (12)(34),

Θ( 14)(23) = (12), Θ(13)(24) = ( 1 4 ) ( 2 3 ) ,

for B. The fact that S4 has a complete mapping now follows from the corollary

of Theorem 1.

Let us now assume that Sn has a complete mapping with n > 3. Then,

Sn+X =Sn + ( 1 , n + 1)Sn + (2, n + 1)S n + + (n t n + 1)Sn ,

= Sn + Sn (1 , n + 1) + Sn (2, n + 1) + + Sn (n9 n + 1) .


Clearly, two cosets (/, n + l)Sn and (k% n + 1 )Sn (j φ. k) being equal would

imply (/, k, n + 1) G Sn and this is impossible.

Now note that

(/, n + 1) (/ + 1, n + 1) Sn = (/, j + 1, n + 1) Sn = (/ + 1, n + 1) Sn

if 1 < / < re - 1. Also, U , n + l ) ( l , n + l)Sn = ( 1 , n + l)Sn.

We now see that the coset representatives of Sn+ γ by Sn satisfy the con-

ditions of Theorem 1 under the obvious mapping S ( l ) = l , S{i)=i + 1 for

2 <. i <. n and S(n + 1) = 2. Hence, S π + X has a complete mapping and our in-

duction is complete.

The corollary follows from Theorem 7 of [ 4 ] ,

It should be pointed out that the coset representat ives used for Sn+ι in the

argument above do not form a group and hence Theorem 1 is sufficiently stronger

than the corollary to be of decided interest.

THEOREM 3. There exists a complete mapping for the alternating group

Λn$ for all n.

Proof. Aι, Ait a n d A3 (the cyclic group of order 3) possess complete

mappings. Hence assume that there exists a complete mapping for An. Then,

rc+ 1 = An + ( 1 , n, n + 1) An + ( 1 , n + 1, n) An + (2, n + 1) ( 1 , n) An

+ (3, 7i + l ) ( l , n)An + . . . + U - 1 , Λ + 1 ) ( 1 , n)An

and the coset representatives are valid for either a right or left coset decom-

position for An+ι by An,

It is a simple, straightforward verification that the permutation S, given by

S ( l ) = l, $ ( 2 ) = 2 , S(3) = 3, S ( ί ) = i + 1 (4 < i < n), S U + D - 4

satisfies the conditions of our Theorem 1. Here we meet a slight difficulty if

n = 3, but it is known [3, p. 422] that there exists a complete mapping for

/14 and we may take n — 4 as the basis for our induction.

3. Groups of order 2". Although it has been indicated in the literature [4]

that the results of this section are known, it seems desirable (and necessary

for completeness) to include the proofs of these results.

LEMMA 1. Let G be a non-abelian group of order 2n and possess a cyclic

546 MARSHALL HALL AND L. J . PAIGE

subgroup of order 2n~ι. Then a complete mapping exists for G.

Proof. It is known [5, p. 120] that G is one of the following groups:

( I ) Generalized Quaternion Group (n > 3) , A 2n~l = 1, B2 =A2n'\ BAB'ι = A'\

(II) Dihedral Group U > 3), A2n'1 = 1, B2 = 1, BAB'1 = A'\

(III) (n > 4 ) , A2n'1 = 1, β 2 = 1, BAB'1 =A 1 + 2 * ' 2 .

(IV) U > 4) , ^ 2 " " 1 = 1, β 2 = l,

In each case, the elements of the group are of the form

Λ D^ \CL — Ό9 1, , z = 1 ; p = 0, l j .

Let us define a mapping Θ as follows: ( let m ~ 2n~ ),

®(Ak)=Ak; A = 0 , l , . . . , m - l ;

Θ ( / 4 * ) - i 4 * I B β ; A = m, m + 1 , . - . , 2 m - l ;

Clearly, Θ is biunique and we will show that it is a complete mapping for

groups I and II. Thus,

Ak >®(Ak) = Ak .Ak = A2k; A = 0 , l , . . . f m - l .

Ak . Θ(Ak) = Ak - Ak'mB = A2k'mB k = m, m + 1, . . •, 2m - 1 .

>2 i . D 2 _ J2" 2

We see that we have a complete mapping if B = 1 or β = A

A slight calculation in the evaluation of Ak B®(AkB), will show that this

mapping is also a complete mapping for the group IV. It is necessary to use the

fact that n > 4.


In order to obtain a complete mapping for group III, we define:

β(Ak > B)=Ak+m; for /c =

* . β) = 4 * δ ; for A; =

The verification that this mapping is a complete mapping for group III is straight-

forward and will be omitted.

This completes the proof of the lemma.

THEOREM 4. Every non-cyclic % group G has a complete mapping.

Proof. This theorem is known to be true for abelian groups [4] , We may use

induction to prove the theorem if G has a normal subgroup K such that K and

G/K are both non-cyclic Corollary 2, Theorem 1).

In view of Lemma 1, we assume that G is a non-abelian group of order 2n

and does not possess a cyclic subgroup of order 2n~ this implies n > 4. If

G contains only one element of order 2, G would have to be the generalized

quaternion group [5, p. 118] contrary to our assumption. Hence G contains an

element of order 2 in its center and another element of order 2. These elements

together generate a four group V.

If V is contained in two distinct maximal subgroups Mi and M2, then Mγ n M2 —

K D V is a normal subgroup of G such that both G/K and K are non-cyclic. In

this case the theorem would follow by induction.

We now suppose that V is contained in a unique maximal subgroup M\. Gt

being non-cyclic, contains another maximal subgroup M2 and if Mγ n M2 is non-

cyclic our induction again applies. Taking Mί n M2 to be cyclic, we see that

Mi is a group of order 2n" containing a cyclic subgroup of order 2n~ and also

the four group V. Thus Mt is of the type II, III or IV of Lemma 1 or possibly

an abelian group with A2*1'2 = 1, B2 = 1, BAB"1 = A. In all cases, Mx n M2 = {A \.

Now let C be any element of M2 not in \A\. Then by the normality of {A \,

C2 = Ar, where r is even since otherwise C would be of order 2n~l and G has no

cyclic subgroup of order 2n~ι. Also C"1 AC ~ Au with u odd.

Now consider the group H ~ \ A2, B }, which is non-cyclic since n > 4. Here,


Λfi = # + HA = # + AH, and

Thus,

G = H +HA+HC + HAG = H + AH + CH + CAH,

where CAH = ACH since

We see that the elements 1, /4, C, AC are two-sided coset representatives for

flinG.

Define

Θ ( l ) = l , ®(A) = C, Θ(C)=AC9&(AC)=A,

and c o m p u t e :

1 . Θ ( l ) / / = 1 .H;

A Θ ( A ) H = A CH

C Θ( C)H = CACH = C C" AC — Ar AUH = /4/Z since r is even, w odd;

4C .ΘUC)//=/lC4tf = C . C ιACH = CAuH = CAH = ACH.

Hence, with these representatives the hypotheses of Theorem 1 are satisfied

and G has a complete mapping.

4. Solvable Groups. The existence of complete mappings for solvable groups

is answered in the following theorems.

THEOREM 5. A finite group G whose Sylow 2-subgroup is cyclic does not

have a complete mapping.

Proof. Let a Sylow 2-subgroup S2 of G be cyclic of order 2m. Then the

automorphisms of S2 are a group of order 2 m - 1 . Hence in G, S2 is in the center

of its normalizer. By a theorem of Burnside [5, p. 139], G has a normal sub-

group K (of odd order) with S2 as its coset representatives. Since G/K = 5 is

cyclic, the derived group G' is contained in K; and clearly,

Π ggee


S is cyclic of order 2m and hence Π s = p, where p is the unique element

of order 2 of S2. Thus, s G S

Π g Ξ p U : ° H P (modK);gGG

and since G'CX, the Corollary of Theorem 1 [4, p. I l l ] is violated and G does

not have a complete mapping.

THEOREM 6. A finite solvable group G whose Sylow 2-subgroup is non-

cyclic has a complete mapping.

Proof. By a theorem of Philip Hall, a solvable group has a p-complement

for every prime p dividing its order. Thus, if S2 is a Sylow 2-subgroup of G

and H is a 2 complement, G — H S and H n S = 1. S has a complete mapping

by Theorem 4 and //, being of odd order, has a complete mapping. By Corollary

1 of Theorem 1, G has a complete mapping.

As further evidence in support of our conjecture we have the following

special theorem.

THEOREM 7. Let G be a finite group whose Sylow 2-subgroup is not cyclic.

If G has (G:S2) Sylow 2-subgroups and the intersection of any two Sylow 2-

subgroups is the identity, G possesses a complete mapping.

Proof. By a well known theorem of Frobenius, G is a factorable group;

that is, G = N S 2 , where N is the normal subgroup consisting of all elements

of odd order. We now apply Corollary 1 of Theorem 1.

REFERENCES

1. P. Bateman, Complete mappings of infinite groups, Amer. Math. Monthly 57 (1950),621-622.

2. R.H. Bruck, Finite Nets, L Numerical Invariants, Can. J. Math. 3 (1951), 94-107.

3. H.B.Mann, The construction of orthogonal latin squares, Ann. Math. Statistics13 (1942), 418-423.

4. L. J. Paige, Complete mappings of finite groups, Pacific J. Math. 1(1951), 111-116.

5. H. Zassenhaus, The theory of groups, Chelsea Publishing Co., New York, NewYork, 1949.

OHIO STATE UNIVERSITY

THE INSTITUTE FOR ADVANCED STUDY AND


RELATIVIZATION AND EXTENSION OF SOLUTIONS

OF IRREFLEXIVE RELATIONS

MOSES RICHARDSON

1. Introduction. Let >- be an irreflexive binary relation defined over a

domain 2) of elements α, 6, c, . We represent the system (5), >-) by an oriented

graph G by regarding the elements of 3 as vertices of G and inserting an arc

ab of the graph, oriented from a to b, if and only if a >- b. The sentence " α >- b"

is read " α dominates 6". A set V of vertices is termed internally satisfactory1

if and only if x G V and γ E V implies x ^j- y. A set V of vertices is termed ex-

ternally satisfactory if and only if γ E 5) — F implies that there exists an % E F

such that % >- y. A set F of vertices is termed a solution of G, or of ( 3 , >~), if

and only if it is both internally and externally satisfactory. In [4], various suf-

ficient conditions for the existence of solutions were established.

By a subsystem Oo,/*") of the system (§>,>-) is meant a system where

5)0 C 5) and the relation >- for the subsystem is merely the restriction of the

relation >- for the supersystem (5), >-). Let Go be the graph of the subsystem

(ô> >"") a n d l e t ô he a solution of Go. A solution V of G is termed an extension

of Vo if Fn ®0 = Vo; in this case VQ is also said to be relativized from V. In

this paper, some sufficient conditions for the existence of relativizations and

extensions of solutions are presented. More elegant and more effective extension

theorems, especially with a view toward possible applications to the theory of

ra-person games, remain to be desired. It is hoped that the present paper may

serve to stimulate interest in this apparently difficult problem.

2. A theorem on relativization. If H is a subgraph of the graph G, then the

graph obtained by adding to H all the arcs of G which join pairs of vertices of

H will be termed the juncture of H (relative to G) and will be denoted by //.

In [ 2 ] , internally satisfactory is called satisfactory with respect to non-domination,and in [4] it is called ^/- -satisfactory.

Received March 1, 1954. Part of the work of this paper was done at the Institute forAdvanced Study in 1952-3, and part while the author was consultant to the LogisticsProject sponsored by the Office of Naval Research in the Department of Mathematics atPrinceton University in 1953-4. A statement of many of the results contained hereinappeared without proofs in L 5 J .

Pacific J. Math. 5 (1955), 551-584

551

552 MOSES RICHARDSON

H is termed a conjunct subgraph of G if and only if H = H.

The graph Go of a subsystem (§) O f >-) oί the system (§),>-) having the

graph G is a conjunct closed subgraph of G. If H is any subgraph of G, proper

or not, and x is any vertex of G, then D~ι(χ9H) shall denote the set of all

vertices y oί H such that y >» x. If v¥ is any set of vertices of G, let

D'\X9H)= U D-H%,«),

and let

D-n(X,H)=D-ι(D-n+\X,H), H)

for Λ > 1. Let D°(X,H) = Z by definition.

THEOREM 1. // Go is α conjunct subgraph of G and V is a solution of G,

then a sufficient condition for V n 5)0 to be a solution of Go, where 5)0 is ίAe

set of vertices of Go, is

(1) D - y ,

Proof. We must prove that F n S)o is both internally and externally satis-

factory with respect to G o . That is we must prove that

( a ) χ9 y E V n S)o implies x >/-y relative to Go, and

(b) y G ? ) 0 - F n 5 ) 0 implies that there exists an % € F π S 0 such that

Λ; >- y relative to Go.

But (a) follows immediately from the facts that GQ is a conjunct subgraph of

G and that V is a solution of G. To prove (b) , consider any y E S)o - Fn §)0.

There exists an Λ; E F such that Λ; >- y relative to G since V is a solution of G.

Then Λ; E D"ι(y9 G) C S)o by hypothesis. Thus x$ y E ®0

a n ( l t n e oriented arc

%y C G. Since GQ is a conjunct subgraph of G, arc xy C G o . This completes the

proof.

REMARK. It would suffice to replace Condition (1) by the weaker condition:

y G S o - Fn 5)0 implies that there exists a vertex x E Fn S)o such that % >- y.

3. An extension theorem. If X C 5), let the predecessor-set of Z relative to

G — Go denote the set

RELATIVIZATION AND EXTENSION OF SOLUTIONS OF IRREFLEXIVE RELATIONS 5 5 3

P(χt G~G0) = U Dmn(X,G-G0).n-i

B y a predecessor-sequence p(xo,G-Go) of x0 E 2)Q r e l a t i v e to G - Go i s

meant a maximal regression2 #o, %i,%2» ' ' •» of finite or infinite length, such

that all its vertices except possibly XQ itself are in G - Go; that is, such that

one vertex xn is chosen from the set D~ι(xn_l9 G - Go) for each n > 0, all

xn's being distinct. Let p* (x0, G — Go ) be the set of all vertices of the pre-

decessor-sequence p (xo9 G — GQ ) other than XQ itself. A predecessor-sequence

is termed trivial if and only if p*(#o» G ~ Go) is empty. We have

for all predecessor-sequences p (x0$ G - Go ) of XQ relative to G - GQ. Note

that the elements of the predecessor-set of x0 or of a predecessor-sequence of

x0 are not necessarily ancestors of XQ, although every ancestor of XQ belongs to

at least one predecessor-sequence of XQ (all relative to G — GQ), If >- is not

asymmetric then a source, which has no ancestor, may have non-trivial pre-

decessor-sequences.

Throughout the sequel we suppose that Go is the graph of a subsystem

(® 0 , >-) of the system (3) f>-) the graph of which is G, that Vo is a given

solution of Go, and that 3

T H E O R E M 2. Suppose that:

(1) All non-trivial predecessor-sequences p (XQ, G — GQ ), XQ E 2>0, are

either infinite or, if finite, of odd length if XQ £$OO and of even length if XQ E F O ;

(2) D( Vo, G)n D- 2 n ( F o, G - Go ) = D ( Vo, G)n D " 2 n + l (f 0 0, G - Go ) = 0 /or

σZZ n > 0;

(3) If h > 0 and k > 0 ore o/ ίAe same parity then

D-h(Vo,G-Go)nD-k(Woo,G-Go) = 0,

and if h > 0 and k > 0 are of different panties then

D-h(V0, G - G0)*D-k{V0, G-Go)= D'h{Woo, G - Go ) n D k(W00, G - GQ) = 0;

2 See [4] for definitions omitted here.3 This is a slight modification of the notation of [ 4 ] .

5 5 4 MOSES RICHARDSON

( 4 ) S _ 3 O C P ( 2 ) O , G - G o ) .

Then a solution V of G which is an extension of VQ exists.

Proof. Let

F = F o u ( u D 2n(V09G-G0)) u ( U D-2n+ι(W009G-G0)),

tf00, G - GoW = Wooυl U D-2n+ι(V0,G-G0)) u( U

We shall show that V is a solution of G. Since Go is a conjunct subgraph of G

and Fo is a solution of Go, it follows that FQ is internally satisfactory relative

to G. By (4), 2) = F u I P . By (3), F n IF = 0; hence W = 3 - F. We have only

to prove:

(a) F n Z ) ( F , G ) = 0 ;

(b) WCD(V9G).

Proof of ( a ) . If Λ; E FQ, y G Fo, then x yf- γ since FQ is internally satis-

factory relative to G.

If x e Fo, y G D " 2 n ( F o, G - Go ), then % )f y by (2) .

If x e Vo, y e D'2n+ι (IFoo, G - Go ), then Λ y. γ by ( 2).

If x eD"2n(V0%G - G o ) , y 6 Fo, then jc^-y; for x >- y would imply that

* G D- ι ( Fo, G ~ Go ) contrary to ( 3).

If x £ D"2n ( F o, G - Go ), y G D " 2 m ( FOf G - Go ), then % y; for % >• y would

imply that XeD'2m'ι( Fo, G - Go ) contrary to (3) .

If xeD-2n(V0,G-G0), y e D - 2 m + ι ( i F o o , G - G o ) , then x Jf y; for % >^ y

would imply that Λ; G D " 2 m ( ! F 0 0 , G - Go ) contrary to (3) .

If % E D - 2 m + 1 ( f F o o , G - G o ) , y ^ o , then % >f y; for % >- y would imply

that x e D" ι ( F o, G - Go ), contrary to (3) .

If % e D - 2 m + l ( t F 0 0 , G - G 0 ) , y £ D - 2 * ( F o , G - G o ) , then x^γ; for x ^ ywould imply that x e D"2n~ι (V0$G - GQ) contrary to (3) .

RELATIVIZATION AND EXTENSION OF SOLUTIONS OF IRREFLEXIVE RELATIONS 555

If x£D-2m+ι(W0Q,G~G0), γeD-2n+ι(W00$G-G0), then x >f y; for

x >- y would imply that x G D"2n (Woθ9 G - Go ) contrary to (3) .

Proof of (b) . If y G IFOO, then there exists an x G Fo such that # >-y.

If y G β - 2 " + 1 ( F o , G - G o ) , then there exists an x G D'2n( VQi G - Go ) such

that Λ; >-y, since y belongs to some predecessor-sequence p(% 0, G - G o ) of

some %0 G Fo and such a predecessor-sequence is infinite or of even length by

(1).

If y eD'2n(W00,G~G0), then there exists an x G D'2n'1 (WOOi G - Go ) such

that x >- y, since y belongs to some predecessor-sequence p(xo9G — Go) of

some %o G $oo* a n c^ such a predecessor-sequence is infinite or of odd length by

(1). This completes the proof.

C O R O L L A R Y . Suppose Conditions ( 1 ) and ( 4 ) of the theorem above, and

that:

( a ) No vertex of any P(XQ9G Λ GO ), XQ G ®O> is adjacent to any vertex of

®o other than XQ; and if XQ and XQ are distinct vertices of ®o then

P(χo,G-Go)nP(xζ,G-Go) = O;

(b) No P(xo9 G — GQ) U ixo), XO G 2>O» contains an odd unoriented cycle.

Then a solution V of G which is an extension of Vo exists.

Proof. We have to show that the hypotheses of the corollary imply those of

the theorem. It will suffice to show that if either ( 2 ) or ( 3 ) are false then

either ( a ) or ( b ) will be violated.

If ( 2 ) were false, there would exist either a vertex

xeD(VOfG)nD'2n(Vθ9G-Go)

or a vertex

yeD(Vo,G)nD'2nU(Woθ9G-.Go).

In either case , the first part of ( a ) or ( b ) is contradicted.

If ( 3 ) were false there would exist either

( i ) a vertex


x e D-Hvl, G - Go ) n D-k(wJ00, G - Go)

with h and k of the same parity or

( i i ) a vertex y such that either

Y e D-Hυί, G - Go )n D-Hvi G - Go)

or

y € D - A K 0 , G - Go ) n Z r f e U 0 0 , G - G 0 )

with h and A; of d i f ferent p a n t i e s .

In C a s e ( i ) , Cond i t i on ( a ) would be v i o l a t e d . In C a s e ( i i ) , ( a ) i m p l i e s

i = /. But then P (vι

Q9 G - Go ) u (v^ ) or P (wι

QQ$ G - Go ) u (u)ι

QQ) would contain

an unoriented cycle of odd length h + k contrary to ( b ) .

4. Sinks and inverse bases. We suppose henceforth that 5) - 2)0 C P (5)0»

G - Go ). If H is any conjunct subgraph of G, and % is a vertex of G, let

C - ι ( % , # ) = U D'n(x9H).n-0

That is, C" (x9H) denotes the set of all vertices y of H which chain-dominate

x by means of a chain all the vertices of which, except possibly x, lie in H,

together with x itself; in symbols

If y G C"1 (xfH) and x E C"1 (y, ff), x j/= y9 then Λ; and y are termed cyclically

related relative to //. If y E C"1 (%,//) but x £ C~ι (y,H) then % is termed a

descendant of y relative to H. A sequence #i , %2$ %3> °f vertices of # is

termed a descending sequence of // if xn+\ is a descendant of xn for all τι

(except the last n if the sequence is finite) and if there exists no vertex y

which is a descendant of all xn. If a vertex x of // has no descendant relative

to // then Cml{xtH) is termed an inverse basic set of A/ and x is termed a sink

of this inverse basic set. A subgraph H is termed descendingly finite if every

descending sequence of H is finite. The same inverse basic set may contain

more than one sink; all sinks of the same inverse basic set are cyclically


related relative to //, and any vertex cyclically related to a sink is a sink of

the same inverse basic set of //.

LEMMA I . 4 If H is descendinglγ finite then every vertex of H belongs to

some inverse basic set of H.

Proof. L e t xι be any v e r t e x of H. E a c h d e s c e n d i n g s e q u e n c e Xι9X29 # 3 , •••

of H b e g i n n i n g with X\ h a s a l a s t e l e m e n t x\. T h e n

- A C V%2» " '9 X 2 ^ V X 3 , Π ) , 9 X\ 1 ^ \ X \ 9 t i )

but

%2 0 C- L ( %i , ), %3 ^ C- l ( %2 , / / ) , . . . , %λ ^ C- [ ( Xλ _ £, / / ) .

Hence

C - 1 ( λ ; 1 , / / ) C C'ι(x2,H) C . C C ^ U ^ t f )

and

λC'ι(xλ,H) = U C'Hxi9H)

is an inverse basic set containing %i of which x\ is a sink.

LEMMA 2. // // is descendinglγ finite, no proper subset B of an inverse

basic set A is an inverse basic set.

Proof. Suppose contrarywise that B were an inverse basic set and a proper

subset of A. Let b be a sink of B and a a sink of A. Then B = C"ι{b9H) and

A - C"1 (α, H). Since B is a proper subset of /4, 6 a and 6 G C'1 (a9H). Since

the sink 6 can have no descendant relative to H9 we have a G C"ι{b9 H)9 other-

wise a would be a descendant of b. Then C"1 (α, //) C C"1 (b9 H), or A C B.

Therefore A = β contrary to hypothesis.

By an inverse basis of // is meant a set S of vertices of H such that (a)

x G S, y G S, x ^ y9 implies that x is not chain-dominated by y relative to H,

4Lemmas 1-5 are duals, in an obvious sense, of Lemmas 1-5 of [4] which are in turngeneralizations of theorems of Kδnig [ 1 , pp. 88-90], for finite graphs. Lemma 2 of [4,p.58l] should be corrected by adding to its statement "if B has a source", and de-leting from the proof all mention of Case (c); this change does not affect the rest of [ 4 ] ,


and (b) y G//n S — S implies that there exists a vertex x of S such that x is

chain-dominated by y relative to H (that is, y G C ' 1 (x$ //)).

LEMMA 3. Every descendinglγ finite subgraph H has an inverse basis.

Proof, Let the distinct inverse basic sets of H be Bί9 B2$ , where βχ ^ βy

for i 5^/. (The range of i and / is any lower segment of ordinal numbers, finite

or not.) By Lemma 1, every vertex of H belongs to at least one β t . Let 6; be a

sink of 5 j . Then no b{ chain-dominates bj9 i ^ j . For, if so, i , G C"1 (bj9 //).

Then bi has 6y as a descendant unless bj G C~ι (bi,H); that is, unless 6j and

6y are cyclically related relative to //. In this case,

C'ί(bhH)cC'ι(bJ9H) and C-ι(bJ9H)CC'ι(bhH);

that is, βj = βy, a contradiction. Let S be the set of b^s just chosen, con-

sisting of one sink from each inverse basic set β;. It has just been shown that

Condition (a) of the definition of inverse basis is satisfied by S. That Con-

dition (b) is satisfied follows immediately from Lemma 1.

LEMMA 4. If H has an inverse basis S and b{ G S9 then C" (b(9H) is an

inverse basic set of which b( is a sink.

Proof. If not, 6t has a descendant p in H. That is,

bieC'Hp.H) but P£C'ι(bi9H).

Since p G H n §>, there exists a vertex bj of S such that p G C" (bj9H). Now,

bj φi b[ since p f. C~ (bι9H) Hence b{ chain-dominates p which chain-dominates

bjf so that bi chain-dominates bj since chain-domination is transitive. This

contradicts the fact that b{ and bj both belong to the inverse basis S.

LEMMA 5. Every inverse basis S of a descendinglγ finite subgraph H con-

sists of one sink from each inverse basic set of H.

Proof. By Lemma 4, each vertex of S is a sink of some inverse basic set.

Two distinct vertices of S cannot both be sinks of the same inverse basic set

since, if so, they would be chain-dominated by each other. There remains only

to show that every inverse basic set has a sink in the given basis S, Suppose

β were an inverse basic set none of the sinks of which were in S. Let b be a

sink of β. Since b is not in S, there exists a vertex b' of S such that b chain-

dominates b\ Hence C"l(b9H) C C~ι{b'9H). But b has no descendant relative


to H s ince b is a sink. Therefore 6 ' and b must be cyclically related relative

to // s ince, if not, b/ would be a descendant of 6. Therefore C" {b'9H)C

C'ι(b9H\ so that C'\b'%h)=C'\b9H) = B. Then b' is a sink of B which does

lie in 5.

5. Progressively finite graphs. A graph H is termed completely descend-

ingly finite if and only if all its closed subgraphs are descendingly finite. A

sequence \xn\ of vertices of H is termed a progression of // if and only if

xn >-#ft + ι, and Cl (xnxn + ι) C H for all n (except the last if the sequence is

finite ). // is termed progressively finite if and only if all the progressions of

// are finite.

LEMMA 6. A necessary and sufficient condition that H be completely de-

scendingly finite is that H be progressively finite.

Proof. If H is progressively finite then it is descendingly finite. If H is

progressively finite then every closed subgraph of H is progressively finite.

Hence if H is progressively finite then it is completely descendingly finite.

If H is completely descendingly finite, there can exist no infinite progression

#ι >~ ^2 >~ y^χn >"•••• For, if so, the subgraph consisting of the vertices

%i and the oriented arcs x^x^x (i - 1, 2, 3, ) would constitute a closed sub-

graph which would not be descendingly finite. This completes the proof.

For example, the graph G of Figure 1 is descendingly finite but not com-

pletely descendingly finite since G — St(y) is an infinite progression.

We suppose henceforth that Cl (G — Go ) is progressively finite^ where Go is

a conjunct closed subgraph of G having the solution Vo. Let5

Woo = D(V0,G0) and Wo = D( Vo, G) υ D'1 ( Vo, G - Go).

Let

G.t = G - S t ( F 0 u f 0 ) .

Let V.i be an inverse basis of G. ι which exists by Lemma 3. For each finite

ordinal number k >_ 1, let

W.k=D(V.k,G.k)uD-HV_k,G_k),

^This is a slight modification of the notation of [4],


ad inf.

Figure 1


and

- St [ U ( Vmi u Wmi ) I = G Λ - S t ( V.-k u »LΛ

and let F_ _ j be an inverse basis of C^-i

LEMMA 7. G.^-i is α conjunct subgraph of G for all k >, 0.

Proof. Any arc of G not in G.^.i lies in

St f U (F. u IF.,.)]

and hence has at least one endpoint in this star. Thus if x and y are vertices

of G_£_i and x >- y relative to G then x >- y relative to G.£. i since arc %y

cannot lie in the star while both endpoints are in G./ . ι

LEMMA 8. For α/Z A; > 0,

u vmi

o < j X /c+i

is internally satisfactory.

Proof. We prove the lemma by mathematical induction.

For k - 0, we must prove that

( F o u F . 1 ) π D ( F o u H l f G ) = 0.

(1) #, y G Fo implies Λ; > - y relative to G; for % >/- y relative to Go since

Vo is a solution of Go and Go is a conjunct subgraph of G.

(2) x E Vo, y € H i implies Λ; )f y relative to G; for Z) ( VOf G - Go ) n G. t = 0

by definition of G. t while V_χ C G. j .

(3) x E F . i j G F o implies % >f y relative to G; for ZTι ( Vo, G - Go) nG.! = 0

by definition of G. i while F. t C G. ι

(4) Λ;, y G F . i implies x >/- y relative to G; for F. t is an inverse basis of

G.i which implies x >/- y relative to G_ t while G_ i is a conjunct subgraph of G

by Lemma 7.

Assuming that U.^ , Vm( is internally satisfactory, we complete the proof by


showing:

( b ) Vmkmln D ( F Φ 1 , G ) = 0 ;

(c ) / U VmλnD(Vmkml9G)=0.\i<k J

If

x e U Vmim y G K.L..1 ,

then % >ί- y; for if % >- y then

y e U

and y jέ G.^ . i , while F-^-i C G.^. i This proves ( a ) . Since G.£. i is a conjunct

subgraph of G, x >- y relative to G, where x, y E Vm/Cmχf would imply x >-y

relative to G./j . i, contrary to the definition of inverse bas i s . This proves ( b ) .

If

then Λ; >/- y; for if x >- y then

%e U D " ι ( H / t G . ^ c U Wmi

so that Λ: G.^.. t , a contradiction. This completes the proof*

It may happen that G_n = 0 for no finite ordinal n, in which case we may let

V = any inverse basis of G , and

W.ω = D( V.ω, G.ω) uD-ι(V.ω,G.ω),

and so on. Transfinite induction shows that if β is an ordinal number for which


PLα is nonempty for all α < β then

U V.aa<β

is internally satisfactory. Let the cardinal number of the set 2) be K . Let λ be

the next largest ordinal after those of 3 ( K ) where 3 ( K ) is the set of all

ordinal numbers of well-ordered sets having cardinal number jr Then no

matter how we well-order the elements of ®, its ordinal number is < λ. Well-

order them as follows:

•• Xa»Xa+l$ '• xβ,Xβ + 1, * * Xy 9Xy +\9

V W

Then every vertex of 2) is in some F.ζ or some ίf.ζ with ζ < λ. Let K be the

lowest ordinal for which G.κ = 0. Then every vertex of G is ultimately used up

in some fiζ or ULζ, ζ < K. We have then the following theorems in which we

let

V= U V.a:0 < α<κ

THEOREM 3. If Vo is a solution of the subsystem (®o>^~) °f t n e system

(2), >•), and if the graph C\(G~G0) is progressively finite, and every vertex

of G - Go is in the predecessor-set P (®o» G - Go), then V is a maximally in-

ternally satisfactory set.

THEOREM 4. //, in addition to the hypotheses of Theorem 3, there exist

inverse bases V.afor each CC with 1 <_ Cί < K such that

D-l{V0,G-G0)CD(V,G) and D" ι ( V.a, G.a) C D( V, G),

then V is a solution of G and an extension of VQ.

THEOREM 5. //, in addition to the hypotheses of Theorem 3, >- is sym-

metric, then V is a solution of G and an extension of VQ.

The proofs of Theorems 4 and 5 are immediate. 6

6 As to Theorem 5, the fact that if >~ is symmetric then every maximally internallysatisfactory set is a solution is established in [2],


THEOREM 6. If the hypotheses of Theorem 2 are satisfied, then so are the

hypotheses of Theorem 4.

Proof. Let

V.i = D-2( Vo, G - Go ) u D - ι ( IF,,,,, G-Go),

V.i = D-3(V0, G - Go ) u D - 2 ( I F 0 0 , G - Go ) = D ' ι ( V.u G - Go) u D( F ^ , G - G o ) ,

F.j =* Z)"4( F o , G - Go ) u D-'dT'oo. G - G o ) ,

and so on. Then

= U F.α

0< α

and

H7= U !F.α = IF0 0uUD-2"+ 1(F0,G-Go)uUD-2 n(IF0 0,G-G0),0 £ α

so that V is a solution.

There remains to show that V.a is an inverse basis of G. α . Clearly, neither

of two distinct vertices x9 y^V.a chain-dominates the other by virtue of the

parity restrictions (2), (3) of Theorem 2. We must show now that every vertex

γ oί G.a chain-dominates some x of K α . This is obvious since by (4) every y

belongs to P ( $ 0 , G - G 0 ) , t h a t i s> t o s o m e D'n(Vθ9G—GQ) or to some D'n(W00,

G - Go), that is, to some H α or W_a. By (1) it is clear that every D"l ( H α , G.α)C

D(VSG). This completes the proof.

The example of Figure 2 shows that Theorem 4 is less restrictive than

Theorem 2. For

but an extension exists and the hypotheses of Theorem 4 are satisfied.

6. Some extension theorems. If H is a subgraph of G, let

K(x,H)=D(x,H) u Z ) - 1 ( % , # ) ,

RELATIVIZAΉON AND EXTENSION OF SOLUTIONS OF IRREFLEXIVE RELAΉONS 565

Figure 2


let

K(X,H)= U Kix.H), X C S ;x£X

let

Kn(X,H)=K(Kn ι(X9H),H) lorn > 1.

That is, Kn(X,H) denotes the set of vertices of H connected to vertices of X

by unoriented one-dimensional chains of length n.

LEMMA 9. If 2) — S)o C P ( Vo$ G - Go ), then every inverse basic set B of

G-j-i has a sink in K2 ( F.j , G./.i), i >_ 0.

Proof, Suppose i = 0. Each sink y of B chain-dominates some vertex of

Vo since 5) - ®0 C P ( Vo, G - Go )• Consider the chains of minimum length m by

which y chain-dominates vertices of FQ. Then m >_ 2 since /£( PQ, G — GQ ) n

G_ι = 0 . Suppose the lemma were false, so that m > 2, and let yQ be a sink of

B for which this minimum length is attained. Then there exist distinct verticesχl$ χ2$ 9 xm- i of G ~ GQ such that

ϊo>- xmΊ >~xm-2 >- •*• >- x l >• v{

for some t>0 G FQ. Then either

(1) Λm-iJfG.i,

or ( 2) x m . i G G. i and is a descendant of yQ,

or (3) %m-i G G β l and is cyclically related to y0 relative to G.χ.

In Case (1), xm.\ £ Vo v WQ so that

xm-i£Kl(V0,Gmi) and yQ eKHV^G.,)

contrary to the supposition that the lemma is false. In Case (2) yQ is not a

sink of B since a sink can have no descendant. In Case (3), m is not the mini-

mum length since xmm\ would be a sink of B which chain-dominates vJ

Q by means

of a chain of length m — 1.

Now suppose i > 0. Let B be an inverse basic set of G_, _i Each sink y of

B chain-dominates some vertex of Vm( since Vmj is an inverse basis of G.t 3 G.t . χ

RELATIVIZAΉON AND EXTENSION OF SOLUTIONS OF IRREFLEXIVE RELATIONS 567

Consider the chains of minimum length m by which y chain-dominates vertices

of Pίj. Then m >_ 2 since K{ V-i, G.j) n G-i-ι ~ 0. Suppose the lemma were false,

so that m > 2, and let yQ be a sink of B for which this minimum length is at-

tained. Then there exist distinct vertices χu x2, 9 xm. i of G_; such that

y o >- %m.i >- Λm.2 > >- xι >- ^ for some i Λ G K.j .

Then either

(1) Xn.iϊG.i.i,

or ( 2) χmm j G G. t _ L and is a descendant of y 0,

or (3) xm-ι G G. t _ i and is cyclically related to yQ relative to G. . 1#

In Case (1),

and hence %m.\ G Kj u W.i and hence %m. i G X l ( PI;, G.j) so that yQ G A 2 ( Kj,

G.j . i ) contrary to our supposition that the lemma is false. In Case (2), y is

not a sink of B since a sink has no descendant. In Case (3), m is not minimal

since xm.\ would be a sink of B which chain-dominates v { by means of a chain

of length m — 1. This completes the proof.

The example of Figure 3 shows that we must take Kn in the unorίented

sense; for here v]γ G P ( F o , G - Go ), in fact υ\ G D'4 (Vo , G - Go ) but υ\<£

D'2 ( Vo, G ~ Go ) although ^ G K2 ( Vo, G - Go )."

A subgraph // of G is termed progressively bounded at the vertex y if all

progressions of H beginning with γ have lengths forming a bounded set of

natural numbers. H is termed progressively bounded if it is progressively bound-

ed at each of its vertices.

LEMMA 10. // S - ® 0 C P ( F o , G - G o ) and if Cl (G-Go) is progressively

bounded then every vertex y of S• - S o i\s arc element of F. t or $ l t for some finite

ordinal i.

Proof. Every vertex y of 2) — 2)0 i s a n element of C" (v^Qi Cl (G — Go )) for

some t;^ G Vo by hypothesis. Consider all progressions of Cl (G - Go ) beginning

with y and ending with elements of Vo. Their lengths have a least upper bound

M(y) by hypothesis. By Lemma 9, we may select inverse bases

V.i.ιCK2(V.i,G-G.i.ι), »>0.


-1

Figure 3


Since V.i is an inverse basis of G_ u there exist progressions starting with y

and ending with elements of F. t unless y is in Vmjc or W^ with k <^ 1. All such

progressions have lengths < M(y) - 2. For if there existed a progression from

y to some v t 6 F_ ι of length > M(y) - 2, there would be a progression from y

to some element of Vo of length > M(γ) since there exists some progression

from v t to some element of Fo and its length must be > 2 because Vm i CG.i

Similarly the lengths of all progressions from y to elements of V_ι must be

<_ M(y) -2i. But this can be >_ 0 for only a finite number of values of i. Hence

there exists a value of i for which y chain-dominates some element of Vm( by

means of a progression of length 0 or 1; that is, y is in either Vmj or ίF.j .

By a relative cycle (of Cl (G - Go ) mod Fo with modulo 2 coefficients) shall

be meant an unoriented one-dimensional chain lying in Cl (G — GQ) except for

its set of boundary vertices (possibly empty; that is, absolute cycles are in-

cluded among the relative cycles ) which lies in Vo.

THEOREM 7. Suppose that Vo is a solution of Go such that'

(1) Cl (G — GQ) is progressively bounded^

(2) each vertex of every K n" (VQ, G — Go ) is dominated by some element

(3 ) Cl ( G — GQ ) contains no relative cycle of odd length;

( 4 ) 3 ) ~ 5 ) o C P ( F o , G - G o ) .

Then there exists a solution V of G which is an extension of Vo.

Proof. Choose V.i as in Lemma 9. To show that V = U Q 1.

L e t

Then

£K2n-ι(viQ,G-G0)


for some / and n by virtue of the way in which the F.^ were chosen. By ( 2 ) , w is

dominated by some vertex x of 5) — 2)0. If x G F, there is no more to prove. If

x e S - F , then

for some k and m. Hence there exists a relative cycle of odd length, contrary to


THEOREM 8. Let V be any maximally internally satisfactory set containing

Fo such that:

(1) every v G V belongs to K2n(V0,G - Go) for some n > 0;

(2) each element of K m" (VQ9 G — GQ)9 for every m > 0, is dominated by

some element of 5) - S o

(3) Cl ( G - Go ) contains no relative cycle of odd length.

Then V is a solution of G.

Proof. Let

γ E ( 5 ) - S 0 ) n ( % - V ) .

We sha l l show that there e x i s t s a n % 6 F such that x >- y. Since V i s maximally

internal ly s a t i s f a c t o r y , F u ( y ) i s not internal ly s a t i s f a c t o r y . Therefore e i ther

( a ) some v >- y, or ( b ) some v -< y. In C a s e ( a ) , there i s no more to prove.

In C a s e ( b ) ,

y eK2n"ι(V0,G-G0) for some τι > 0.

By ( 2 ) , there exists an x G 5) — S o such that a; >- y. If x G F f there is no more

to prove. If not, that is if x G (2) - ® 0 ) n ( S - F ) , then F u (%) is not internally

satisfactory. Therefore there exists a v G F such that either Λ; >- v or x -< t>.

In either case,

xeK2mml(Vθ9G-Go)

for some natural number m But this together with

y£Kanml{V0,G-G0)


and x >- y imply that there exists a relative cycle of odd length contrary to


COROLLARY. The hypotheses of Theorem 8 imply that

V0,G-G0) and W = §-V =

Proof. We have

Vcυκ2n(V0,G~G0)=E,

and

W = 2> - V C U K 2 m ' ( Vo, G - Go ) = Ω.

Furthermore

K2n ( Vo, G - Go ) n K2m-ι ( Fo, G - Go ) = 0 ,

for, if not, there would exist a relative cycle of odd length. Thus we have

£ n Ω = 0 , E u Ω = S , F c £ , W'Cίl, F u l F = S),

This implies E - V9 Ω = W as follows. Let e E £. Then e E 5) which implies that

either e E F or e E IF. But e E ίP would imply that e E Ω contrary to £ n Ω = 0.

Therefore e 6 F , Hence E C V and therefore E = V. Similarly Ω C IF and hence

Ω = IF. This completes the proof.

Thus Theorem 8 resembles Theorem 2, except that now the parity restric-

tions are on the unoriented chains rather than on the oriented ones, and we do

not restrict the sets Kn(WOOf G -Go).

The examples of Figures 4-6 are covered by Theorem 8 but not by Theorem

2. In Figure 4,

w I e D ( Vo, G) n D -1 (Wo 0, G - G 0 ) £ 0

violating hypothesis 2b of Theorem 2, but the extension exists under Theorem

8. In Figure 5,

v _\ G D'2 (Wo o, G - G o ) n D" ι (Wo 0 , G - G 0 ) ^ 0

violating the second part of the hypothesis 3b of Theorem 2, but the extension


Figure 4


Figure 5


W- 1

Figure 6


exists under Theorem 8. Note also that an odd relative cycle exists mod Go but

not mod VQ, In Figure 6,

v\ eD-2(V0,G-G0)nD-2(W00,G-GQ) ^ 0

and

j2£D-l(V0,G-G0)nD-l(Woo,G-G0)J-0Wn

both violating hypothesis 3a of Theorem 2, but the extension exists under

Theorem 8.

Let μ (X9 G - Go ) denote the set of vertices of G - Go connected to X by

an unoriented chain of minimal length h, where X C 2). Then

μh(X, G - Go ) n μk (X, G - Go ) = 0 for h £ k.

By μ° (X9 G - Go ) is meant Z.

THEOREM 9. Lei F o be a solution of Go where Go i's α conjunct subgraph

of G. Let WQQ = ®o — o β ^ suppose that every vertex of 3 — 5)0 Ϊ'S connected

to 5)0 ^y some unoriented chain. Let

= U μ 2 " ( J / 0 , G - G 0 ) u Un=o m = i

F = U J u 2 ί l - 1 ( F 0 , G - G 0 ) u U μ 2 m ( I F o o > G - G o ) .

w = l m = o

Suppose that:

(1) every element of W is dominated by some element of V;

(2) μh(V0,G-G0)nμk{W00,G-Go)=0

if h and k have the same parity,

(3) no two elements of the same μ2n"1 (tFOo, G - Go ) are adjacent;

(4) no two elements of the same μ n{ VQ, G — GQ ) are adjacent.

Then V is a solution of G which is an extension of Vo .


Prόυf. Clearly 5) = V u Ψ and WCD{V9G). Also (2) implies V n IF = 0.

There remains only to prove that no two elements of V are adjacent.

If *, y E μ2n{V0,G- Go ) then x >f y by (4).

If *, y £ μ2m-1 (iF0o, G - Go ) then x + y by (3).

Let

^ e μ 2 n ( 7 0 , G - G 0 ) , y£μ2m(V0,G-G0), mfn.

Suppose m > n. If x and y were adjacent then y & K2n ι (V0,G — Go). But

2re + 1 < 2m, contradicting the minimal property of μ2m( Vo, G — Go ). A similar

proof is obtained if m < n.

If

* e μ a B - | ( I P o o , G - G o ) , y £ μ a m " ι ( l P o o , G - G o ) , m Φ n ,

then Λ: and y are proved non-adjacent as in the preceding paragraph.

Let

χeμ2n(Vo,G-Go),y£ μ2P ι(W00,G - Go )

and suppose x were adjacent to y. Then

x&K2P(W00,G-G0) or x£K2P-2(W00,G-G0).

Since Λ: is connected to Woo, it is minimally connected to Woo. That is, either

(a) * € μ 2 Λ ( l F O o , G - G o )

or

(b) x e μ

2 h ι{WQ0iG~G0)

for some A. In Case ( a ) , Condition ( 2 ) would be violated. In Case ( b ) , h = p

since either h p would violate the minimal property of some μ.

But h =p contradicts Condition ( 3 ) . This completes the proof.

THEOREM 10. Let VQ be a solution of a conjunct subgraph Go of G such

that every vertex of 5) - ®o *5 connected to Vo by some unoriented chain. Let:


(1) no two elements of the same μ2ι( Vθ9 G — Go), i > 0, be adjacent;

(2) x £ μ ι" (Vo$ G - Go) imply that there exists a j >_0 such that

x -< y for some γ € μ2} ( Vo, G - Go).

Then

V' = U μ2i(V0,G-G0)ϊ = 0

is a solution of G which is an extension of Vo.

Proof. Every element of 5) — 2)0 not in V must be in

IF= U ^ 2 i - l ( F 0 f G - G 0 ) .

Clearly

5) = F u IF and Fn W = 0 .

Also (2) implies W C D (VfG). There remains only to prove that V is internally

satisfactory.

Let

x e μ

2i( Vo, G - Go ), y £ μ2H Vo$ G - G o ) , i ί j .

Suppose i < j . If x were adjacent to y, then y £ K2i ι ( Vo, G - Go ). But 2i + 1 <

2/, contradicting the minimal property of μ2Π VOf G - G o ) . A similar proof

holds if i > j .

Let x9 y G μ ι (Vo$ G - Go). If i > 0, (1) implies that x and y are non-

adjacent. For i = 0, this follows from the facts that Vo is a solution of Go and

that Go is a conjunct subgraph of G. This completes the proof.

The conditions of Theorem 10 do not prohibit entirely the existence in

Cl (G — Go) of adjacent vertices of IF, of odd unoriented cycles, or of transitive

triples. For example, the graph in Figure 7 permits an extension by Theorem 10

and includes the three cited phenomena. Theorems 7-10 may be regarded as

variants of Theorem 2.

7. Dual and alternating procedures. Let G t be a conjunct subgraph of G.


-1

Figure 7

RELATIVIZATΊON AND EXTENSION OF SOLUTIONS OF IRREFLEXIVE RELAΉONS 579

If x £ 5), let D(x$ G — Gι) denote the set of all vertices γ of G - G\ such that

x >- y. If X C 2), let

,G~G 1 )= U D C ^ G - d ) .

Λ G X

For n > 1, let

By the successor-set of /? relative to G - Gι is meant the set

oo

SiX G-G^- U D^ί^G-Gx).

THEOREM 11. Let Gγ be a conjunct subgraph of G, Vγ a solution of G\9

Wί = ®! - Fi where ® t = S) n G t . Suppose that:

( 1 ) for et>ery τι > 0,

F x n Z) 2 n + r ( Vt, G - G r ) = n D2 n (IF ι , G - G % ) = 0

( 2 ) if h > 0 αrac? A; > 0 are of the same parity^ then

Dh(Vι,G-G1)nDk(Wι,G-Gι) = 0;

if h > 0 and k > 0 are of different parities then

Dh(Vι,G-Gι)r\Dk{Vι,G-Gι)=Dh(Wι,G-Gι)nDk(Wι,G-Gι)=O;

(3) S - S 1 C S ( S 1 , G - C ) .

Γλerc there exists a solution V of G which is an extension of V\.

Proof. Let

= F , u U D2n{VltG-Gι)u U D2m ι(Wu G - Gι).n = l m = l


We must show:

( a ) F n D ( F , G ) = 0 ;

(b) 5 > - F c D ( F , G ) .

(a) If x G F j , γ G Vx, then x >jί- y since Vι is internally satisfactory rela-

tive to G.

If x£Vl3 y£D2n{VuG-Gι) thens . j f y; for* >- y would imply D ( F,, G) n

D a B ( ί Ί . G - G ι ) ^ 0 contrary to (1) .

If xG Fi , y e D ^ - ' d ί Ί . G - G ! ) then * y- y; for * X y would imply D ( Vι,

G)n D2m ι(WuG -Gι) ^ 0 contrary to (1) .

If xeD2n(VltG-Gι\ y€Vi, then * )/- y; for * >^ y would imply γ£

D^+HVuG-d) contrary t o ( l ) .

If * e D a n ( ^ l f G - G i ) , y e J 9 2 m ( f 1 , G - G l ) then c ^ y ; for * > y would

imply y e D2n+ι (Vlt G - G t ) contrary to (2) .

If x e D 2 " ( Ft, G - d ), y e D ^ ^ (Wt, G - G t ) then x Jf y; for * V y would

imply y e D 2 n + 1 ( F 1 , G - G x ) contrary to (2) .

If xeD2mml(WltG~Gι), y G F t then a; )f y; for * > - y would imply y G

D ^ d f Ί . G - G t ) contrary t o ( l ) .

If x € D2"1"1 (IΓX, G - Gi ), y G Z) 2 n ( Vx, G - G t ) then x ^- y; for * >• y would

imply y G D 2 m ( I F 1 , G - G 1 ) contrary to (2) .

If % G D 2 m - l ( l F 1 , G - G 1 ) , yeD^-^W^G-Gi) then x >/-γ; for x >- y

would implyy G D2m(Wlt G - Gγ) contrary to (2) .

(b) Let

ι U U D 2 n - 1 ( F 1 , G - G 1 ) υ U D 2 m ( I F i , G - G t ) .n=l m = l

By (3) ,

By (1) and (2), V n W = 0. Hence IF = 2) - F.

If y G IFi, then there exists an x G Vι such that x >- y.

If yG D 2 n - 1 ( F 1 , G - G 1 ) then there exists an x G Vγ u D 2 π ( F t , G - Gx

such that x >- y.

RELATIVΪZATION AND EXTENSION OF SOLUTIONS OF IRREFLEXIVE RELATIONS 5 8 1

If y £D2m(Wl9G~Gι), then there exis t s an x G D2m"1 ( Wx, G - Gι) such

that x >- y. This completes the proof.

COROLLARY. Let Gx be a conjunct subgraph of G, Vι a solution of Gx.

Suppose that:

( a ) no vertex of any S (xι,G - Gχ)9 x\ £ ® i , is adjacent to any other

vertex of 2)χ; and if X\and xâre any two distinct vertices ofî then

( b ) no

S(xlt G - Gι) u (xι), xχ£^>ι,

contains an unoriented cycle of odd length;

( c ) S-^CSI^G-G,).

Then there exists a solution of G which is an extension of V\.

Proof. C o n d i t i o n ( c ) i s i d e n t i c a l with ( 3 ) of the theorem. We h a v e only to

s h o w t h a t ( a ) and ( b ) imply ( 1 ) and ( 2 ) ; t h a t i s , t h a t if e i t h e r ( 1 ) or ( 2 ) were

f a l s e t h e n ( a ) or ( b ) would be v i o l a t e d .

If ( 1 ) were fa l se t h e r e would e x i s t e i t h e r

( i ) a v e r t e x x € D ( F 1 , G ) n D 2 " ( F ι , G - G ι ) ,

or ( i i ) a v e r t e x r G D ( K ι $ G ) n D 2 " - ι ( l F ι , G ~ G ι ),

or ( i i i ) a v e r t e x z G Vx n D 2 n + 1 ( V,, G - Gt),

or ( i v ) a v e r t e x u € Vx n D2n(WuG - Gx).

In C a s e ( i )

xeS{v[9G-Gx)r\S{v{tG-Gx)

and by ( a ) , i - j . But then there e x i s t s an unor ien ted cyc l e of odd leng th in

S(vι

ι<$ G - G\ ) u {v[ ) contrary to ( b ) . In C a s e ( i i ) , the s e c o n d p a r t of ( a )

i s c o n t r a d i c t e d . In C a s e s ( i i i ) and ( i v ) , the first p a r t of ( a ) i s c o n t r a d i c t e d .

If ( 2 ) were f a l s e , t h e r e would e x i s t e i t h e r


( i ) a vertex

x £ Dh ( Vx, G - G ι ) n D k (W ί, G - G t )

for some h9 k of the same parity,

or ( i i) a vertex

y EDh(v[9G - Gv)n Dk{υ{9G - Gt)

for some A, A; of different parities,

or (i i i) a vertex

zeDh(w[$G-Gι)nDk(w{fG-Gι)

for some ^, A; of different par i t ies . In Case ( i ) , the second part of ( a ) is con-

tradicted. In C a s e s ( i i ) and ( i i i ) , ( a ) implies i -j and then ( b ) is contradicted.

Now suppose Go is a nonempty conjunct subgraph of Gι and let Vo be a

solution of GQ. For each natural number n, let £271-1 be constructed by adjoining

to G2n-2 the vert ices of P (5)2n-2$ G - ^2^-2)* where S j = S) n G t , and taking

the juncture; that is ,

£2/2-1 " ^2n-2 U P ( >2n-2f G - G 2 Λ - 2 )

Similarly let

G2n = G2n-ι u 5(5)2n-U^ ~ £2^-1).

Then each Gf is a conjunct subgraph of G + i For x% y G 5)j , # >- y relative to

Gj+i implies Λ; >- y relative to Gz since at least one endpoint of every arc in

Gj+ t - Gj is not in G;.

If GQ intersects every component of G, then

3 = U 5)f .

For then every vertex of G is joined to some vertex of Go by a finite unoriented

chain and therefore lies in some G;. In particular, this is true if G is connected.

THEOREM 12. Let Go be a conjunct subgraph of G which intersects every


£35#~

3 3 3

^ 3 2 ^34 #36

3 2 2

5 23 6 2 l Ί l

^ * "

< f <6 14

' 1 3

Figure 8


component of G, let Vo be a solution of G o , and let Gι9 i >_ 1, be defined as

above. Suppose that for every even i, G( satisfies Conditions ( 1 ) , ( 2 ) , ( 3 ) of

Theorem 2 relative7 to Gj+ l 9 and that for every odd i9 G t satisfies Conditions

( 1 ) and ( 2 ) of Theorem 11 relative to G + i. Then there exists a solution of G

which is an extension of Vo.

Proof. The solution Vo of Go can be extended stepwise to a solution Vγ of

£i> ^2 °f ^29 ' ' '» Vi °f £/» by Theorems 2 and 11 applied alternately. Hence

U°t.o Vι i s a solution of G.

For example, in Figure 8, G; has the set of vert ices 2); = [g.j,, g 2 , g 3 , ].

Then

F l = [ £ l 2 , S l 4 » - - ] u F 0 , ^2 = [ g 2 1 , g 2 5 , ••• i § 2 4 ^ 2 8 > # ] U F 1 »

Theorem 11 is a sort of dual to Theorem 2. Theorem 12 merely uses the

procedures of Theorems 2 and 11 in alternation. Similar processess dual to those

of other preceding theorems can be introduced so as to yield extensions in the

direction of successor-sets rather than predecessor-sets, and similar alternating

procedures can then be used.

T h a t i s , w i t h G(+χ in t h e r o l e of G in T h e o r e m 2 .

REFERENCES

1. D. Kδnig, Theorie der endlichen und unendlichen Graphen, Leipzig, 1936.

2. J. von Neumann and O. Morgenstern, Theory of games and economic behavior,Princeton 1944; 2nd edition 1947.

3. M. Richardson, On weakly ordered systems, Bull. Amer. Math. Soc. 52 (1946),113-116.

4. , Solutions of irreflexive relations, Ann. of Math. 58 (1953), 573-590.

5. , Extension theorems for solutions of irreflexive relations, Proc. Nat.Acad. Sci. 39 (1953), 649-655.

BROOKLYN COLLEGE

AN INEQUALITY FOR SETS OF INTEGERS

PETER SCHERK

Small italics denote nonnegative integers. Let A = {a ί, B = { b ! , be sets

of such integers. Define A + B = 1 a + b \ and put

Thus

/4 ( n ) = /I (0, n ) and A(mtn) ** A(n) —A ( m ) if m < n.

The following estimate is well known:

LEMMA. If m < k< n, n fi A + B, then

(1) k-m>_A(n-k-l, n-m-D +B(m9 k).

Proof. If b = rc —α, then n = α + !) E/l + β . Hence the 4 (/z. — Λ; — 1, n — m — 1)

numbers rc — α with m < n —a <^k and the B (m9 k) numbers b satisfying m < b <_ k

are mutually distinct. The right hand term of (1) gives their total number. It is

not greater than the number k — m of all the integers z with m < z <_ k.

The most important result on A + B is due to Mann [2] : Let n fc C = A -f β.

Then there exists an m satisfying 0 < m < n and n — m ft. C such that

C{m9n) Âin-m-D + Bίn-m-D.

I wish to prove a less well known inequality which is implicitly contained

in [4] and in a paper by Mann [3], The present proof uses an idea by Besicovitch

and is rather simpler than Mann's method [cf. l ]

THEOREM 1. Let

(2) xeA ( % = 0 , l , 2 , . . .,A; h > 0 ) ,

Received December 29, 1953.Pacific J. Math. 5 (1955), 585-587

585

586 PETER SCHERK

( 3 ) 0 G B or 1 G B9

( 4 ) AΪBCC, n £ C.

Finally let

(5) C(n) < AU-Ό + BU).

Then there is an m satisfying

(6) m £ C9 0 < m < n - h - I

such that

( 7 ) C(m9n) >_A(n-m-l) +B(m,n).

We note that ( 7 ) is trivial but useless without the second half of ( 6 ) .

Obviously, ( 2 ) - ( 4 ) imply m > h if 0 G B and m > h + 1 if 1 G B.

Proof. Instead of ( 3 ) , we merely use the weaker assumption that B is not

empty. Let 60 denote the largest b <^n. Thus B(bo9n) = 0. Since C contains

the integers 60 + a with 0 < a <^ n — bo, we have

(8) C(bo,n) >_A(n-b0) > A (n - bQ - 1) = A U - b0 - 1) + B (bθ9 n).

From ( 5 ) and ( 8 ) , b0 > 0. By ( 2 ) , the numbers bOf b0 + 1, , b0 + h lie in

A + B C C. Hence n jέ C implies 6 0 <. τι - A - 1. Thus

( 9 ) 0 < 60 < / ι - A - l .

By ( 2 ) , b0 £C. Let m denote the greatest z < b0 with z j£ C. If no such

z exists, put m = 0. Applying ( 1 ) with k = &o> w e obtain

(10) C ( m 9 b o ) = b 0 - m > A ( n - b 0 - 1 , n - m - 1 ) + B ( m 9 b 0 ) .

Adding ( 8 ) and (10) , we obtain

C(m9 b0) + C(bθ9n) >_ A (n - b0 -~ I) + A {n - bo - I, n - m - I)

+ B(m9 b0 ) + B(bθ9n)f

that i s (7). By ( 7 ) and ( 5 ) , m > 0. Hence m fi C. Final ly ( 9 ) and m < b0 imply

m < n - h — 1.

The following corollary of Theorem 1 was proved in a different way by Mann.

AN INEQUALITY FOR SETS OF INTEGERS 587

T H E O R E M 2. Suppose the sets A, B, C satisfy the assumptions ( 2 ) - ( 4 ) .

Let 0 < 0Ci < 1 and

(11) Λ{x) > ux{x + 1 ) U = A + 1, A + 2 , . . . , Λ ) .

( 1 2 ) C U )

Proof . By ( 2 ) , OeΛ. F u r t h e r m o r e , ( 1 1 ) and ( 2 ) imply ISA. H e n c e , ( 3 )

i m p l i e s 1 £ C. T h u s our t h e o r e m i s t r u e for n = 1. S u p p o s e i t i s p r o v e d up to

τ ι - 1 > 1.

If C(n) >_A(n-l) + B(n), t h e n ( 1 1 ) with x = n~l y i e l d s ( 1 2 ) . T h u s we

may a s s u m e ( 5 ) . C h o o s e m a c c o r d i n g to T h e o r e m 1. By ( 6 ) , n — m — 1 >_ A + 1.

H e n c e , by ( 7 ) , ( 1 1 ) , a n d our i n d u c t i o n a s s u m p t i o n

C(n) >_C{m)+A{n-m~l)+B(m9n)

>_C(m) + α t (7i - m ) + B(m9n)

The case h = 0 of Theorem 2 is due to Besicovitch [ 1 ]. Obviously, this

theorem can be extended to the case that 0 j£ B$ B in) > 0.

A recent result by Stalley also follows readily from Theorem 1.

REFERENCES

1. A. S. Besicovitch, On the density of the sum of two sequences of integers, J.

London Math. Soc. 10 (1935), 246-248.

2. H. B. Mann, A proof of the fundamental theorem on the density of sums of sets

of positive integers, Ann. of Math. 43 (1942), 523-527.

3. , On the number of integers in the sum of two sets of positive integers,

Pacific J. Math. 1 (1951), 249-253.

4. P. Scherk, Bemerkungen zu einer Note von Besicovitch, J. London Math. Soc.

14 (1939), 185-192.

5. R.D. Stalley, A modified Schnirelmann density, Pacific J. Math. 5(1955), 119-124.

UNIVERSITY OF SASKATCHEWAN

ON INFINITE GROUPS

W. R. S C O T T

1. Introduction. Several disconnected theorems on infinite groups will be

given in this paper. In V 2, a generalization of Poincare"s theorem on the index

of the intersection of two subgroups is proved. Other theorems on indices are

given. In § 3 , the theorem [ 3 , Lemma 1 and Corollary l ] that the layer of ele-

ments of infinite order in a group G has order 0 or o(G) is generalized to the

case where the order is taken with respect to a subgroup. In v 4 , it is shown that

the subgroup K of an infinite group G as defined in [ 3 ] is overcharacterist ic

[ 2 ] . In § 5 , characterizations are obtained for those Abelian groups G, all of

whose subgroups H (factor groups G/H) of order equal to o{G) are isomorphic

to G (in this connection, compare with [ 7 ] ) . Again the Abelian groups, all of

whose order preserving endomorphisms are onto, are found ( s e e [ 6 ] ) .

2. Index theorems. If // is a subgroup of G, let i(H) denote the index of

H in G. The cardinal of a se t S will be denoted by o(S).

THEOREM 1. Let Ha be a subgroup of G, α E S. Then

£(Π//α) <Πi(Ha).

Proof.

gtg'2l£ Γi Ha

if and only if

' e f f α for all α G S.

Thus each coset of Π//α is the intersection of a collection of sets consisting

of one coset of Ha for each (X, and the conclusion follows.

COROLLARY 1. (Poincare) The intersection of a finite number of sub-

groups of finite index is again of finite index.

Received December 23, 1953.

Pacific J. Math. 5 (1955), 589-598

589

590 w. R. SCOTT

C O R O L L A R Y 2. // i(H)=B9 then G has a normal subgroup K such that

<BB.

Proof. Let N(H) denote the normalizer of H, and Cί(H) the conjugate

class of H. Then

HCN(H), o(Cl(H))=i{N{H)) <B.

Thus if K is the intersection of the conjugates of //, Theorem 1 gives i(K) < B .

REMARKS. For every infinite cardinal A9 there is a simple group G of order

A (for example, the *'alternating" group on A symbols). Thus G has no sub-

groups of index less than or equal to B if 2 < A. In particular, if A is such

that B < A implies 2 < A9 then G has no subgroup of index less than its

order A, This is in sharp contrast to the behaviour of Abelian groups, which

have 2A subgroups of index B for Ko <_ B <_ A9 A > Xo [4] , It is an unsolved

problem as to whether there exists a group G of order A with no subgroups of

order A, for A > Ko.

Let U denote the point set union, and + and Σ direct sums ( the latt ice union

of subgroups will not be u s e d ) . If Γ is a nonempty subset of a group G, let

iR( T) = min o (S) such that \JTxa ~ G9 C ί E S .

Define iL(T) similarly, and let i(T) be the smaller of iR(T) and iî T),

T H E O R E M 2 . If Hι9 ϊ = 1 , * ,n9 are subgroups of G such that i ( / / j ) > ^

A > Ko, then i ( U ^ ) > A.

Proof. The theorem is true for n = 1. Induction on n If, contrary to the

theorem, i(UHi) < A, then, say,

( n

U Hi

- „ i=i

with o ( 5 ) < /4. S i n c e ί(Hι) >_ A9 t h e r e e x i s t s a n % 6 G s u c h t h a t

is empty. Hence

ON INFINITE GROUPS 591

Therefore

n n I n \UfyC U Hi(e υ(ϋaxax'ι)) = U U HΛxβ,

i = i *=2 / 3 G s ' \ i = 2 /

where o ( S ' ) < /4. Hence

= U ( U / / Λ * α = U U U Hixnxa= U U/ αCS / 3 £ s ' j=2 P G "

This contradicts the induction hypothesis. Hence the theorem is true.

REMARK. For every infinite cardinal A, there is a group G of order A9 con-

taining an increasing sequence { Hn } of subgroups, each of index A9 such that

m n «= G.

Let I/A = 0 for A > Ko.

THEOREM 3. // Hi is a proper subgroup of G9 (i - 1, , n) and Σ 1/i (Hi )<

1, then UHt £ G.

Proof. L e t H\, , HΓ have finite index, the others infinite index (if r = 0,

the theorem follows immediately from Theorem 2 ) . L e t

D = Π //; .1

Then D has finite index in G, and it is well known that (UΓ Hi) n Dx is empty

for some x G G. Hence, if U? fff = G, then Dz C U ^ j ffίf whence U ^ + 1 ^ has

finite " index" in contradiction to Theorem 2. Therefore U" Hi ^ G.

3. Layers. Let T be a subset of G, and let n be a positive integer. Let

L i n , T ) = \ g \ g n £ T , g

r £ T f o r 0 < r < n \ ,

Moo, 7') = U | g Λ ( ί 7 ' , n = l , 2 , . . . } .

For T - e, the L(n, Γ) have been called layers. The following theorem general-

izes [3, Lemma 1 ]•

THEOREM 4. Lei G be an infinite group9 H a subgroup9 P a set of primes

and

592 w. R. S C O T T

U U L{λp,H)\\iL(ω,H).,£P λ /

Proof. Deny the theorem. Let x E S. If XeL(λp9H) then xλeL(p9 H).

Hence we may assume that # E L (oo, //) or x E L (p, H), p E P.

Case 1. o (/V(%)) = o ( G ) , where N(%) is the normalizer of %. Then o {N(x)~

S) = o ( G ) . If y E /V (% ) - S, then yΓ E // for some r such that (r, p ) = 1 (if p

e x i s t s ) . If xγ £ S then also (xy)n £ H for some n such that (n 9 p) = 1 (if p

e x i s t s ) . Thus

and # Γ / Ϊ E //. B u t (rn9 p ) = 1 if p e x i s t s , a n d , in any c a s e , we h a v e a c o n t r a -

d i c t i o n . H e n c e xγ E S a n d

o ( S ) > o ( % ( / V U ) - S ) ) = o ( / V U ) - - S ) = o ( G ) ,

a c o n t r a d i c t i o n .

C a s e 2. o ( / V U ) ) < o ( G ) . T h e n o ( C Z ( % ) ) = o ( G ) .

C a s e 2 . 1 . o(H) = o(G). T h e n o ( G ) r i g h t c o s e t s of N(x) i n t e r s e c t H,

T h u s t h e r e are o(G) e l e m e n t s of t h e form h~l xh. But if {h"lxh)n E H t h e n

xn£H, w h e n c e n = λ p and A" 1 xh £ S. T h e r e f o r e o ( 5 ) = o ( G ) , a c o n t r a d i c t i o n .

Case 2.2, o ( / / ) < o ( G ) . We h a v e , s i n c e o{S) < o(G),

(1) o(G)=o(Cl(x))= 21 o ( C Z U ) n L U , t f ) ) .(τι,p) = l

If o ( G ) = Ko, and o{x) = oo, then since H is finite,

C Z U ) C L ( o o , / / ) C S,

a contradiction. If o ( G ) = K0, and o(x)~m, then Cl(x) n L(n, H) is empty

for n > m. Hence, by ( 1 ) , there exis ts , regardless of the s ize of o ( G ) , an n

such that (n9 p ) = 1 and

o(Cl(x)nL(n9 H)) > o(H)o{S).

Let


AU9T) = { g \ g

n e T \ .

T h e n A(n9H) D_L(ntH), h e n c e

o(Cl(x)t>A(n,H))= Σho(CUx)nA(n,h)) > o(H)o(S).

Hence there exists an ho £ H such that

o{Cl{x)nA{n,h0) > o(S).

There i s then a b 6 G such t h a t (b"1 xb)n = h0, whence

xeCl{x)nA(n9 bhob'1).

If

q eCl(x)n A(nibhQb-1),

then

q

n = bhob l =xn.

Hence if qr G H9 then

xnr *qnΓ £H

and p I ΓΪΓJ whence p | r. Thus <7 G S in any case. We have

o(S) >_o(Cl(x)<\A(nsbh0b"l))=o(b{Cl(x)<\ A(n,ho))b-1)

= o(Cl(x)nA(n,h0)) > o(S).

This contradiction shows that the theorem is true.

C O R O L L A R Y . // H is a subgroup of the group G9 then o ( L ( o o , / / ) ) - o{G)

orO.

Proof, In Theorem 4, let P be the empty set.

4. An over-characteristic subgroup. Neumann and Neumann [ 2 ] have defined

a subgroup K of G to be over-characteristic in G if and only if ( i ) K i s normal,

and ( i i ) G/K £ G/H impl ies KCH.

594 w. R. SCOTT

Define ( s e e [ 3 ] ) a subgroup K of an infinite group G a s follows. Let E(x)

be the set of g G G such that x is not in the subgroup generated by g, and let

K be the set of x G G such that o(E(x)) < o ( G ) .

THEOREM 5. //"G is infinite$ and K is defined as above, then K is an over-

characteristic subgroup of G.

Proof, ( i ) K is normal since it is fully characteristic [3, Theorem 6],

(i i) Let G/K ^ G/H.

Case 1. K is finite. Then [3, Corollary 3 to Theorem 8]

K2 =K(G/K) =e.

Hence K(G/H)=e. Now

o(G/H)=o(G/K)=o(G).

If there exis ts a k G K — H, then

o ( E ( k H ) ) < o ( E ( k ) ) < o ( G ) = o ( G / H ) .

Hence kH G K(G/H). This is a contradiction. Hence K C^H, and X is over-

characteristic.

Case 2. X is infinite. Then [3, Theorem 5] A is a p°° group, and [3, Theo-

rem 8] G/K is finite. If there exists a k G X - H then

implies k' £ K - H, and

This contradicts the finiteness of G/H. Therefore X C//, and since G/K is

finite, K ~ H. Hence K is over-characteristic.

5. Abeliaπ groups with special properties.1 If G is an Abelian group such

that 0 C H C G implies G ~ H for subgroups //, then it i s trivial that G i s 0 or

cyclic of prime or infinite order, and conversely. This naturally leads to the

problem of finding those groups which p o s s e s s the following property:

1For the facts used without proof in this section, see [ l ] .


(Pi) G i s Abel ian, and if // i s a subgroup of G s u c h that o(H) = o(G) then

G ~H.

THEOREM 6. G has property (Pi) if and only if (i) G is finite Abelian,

( i i ) G is a p°° group, ( i i i ) G is a direct sum of cyclic groups of order p, p a

fixed prime, ( i v ) G is infinite cyclic, or ( v ) G is the direct sum of a non

denumerable number of infinite cyclic groups.

Proof. If G is of one of the above five types, then it is either trivial or

well-known that G has property ( P i ) .

Conversely, suppose that G is infinite and has property ( P i ) . Let T be the

torsion subgroup of G.

Case 1. o(T) < o(G). Then (see, for example, [3, proof of Theorem 9,

Case 1]) there is a free Abelian subgroup H of G such that o(H) = o ( G ) .

Hence G 21 H. If the rank of G is non-denumerable, we are done. If the rank of

G is countable, then G is countable and contains an infinite cyclic subgroup.

By (Pi ) , G is infinite cyclic.

Case 2. o(T) = o ( G ) . Then G 21 T, that is, G is periodic. If Gp is a non-

zero p-component of G, then G = Gp + Hp, hence G 21 Gp or G 21 //p , a con-

tradiction unless //p = 0. Hence G is a p-group. Thus G = D + R, where D is a

divisible (that is, nD — Ό) and R a reduced (no divisible non-zero subgroups)

p-group. Hence G 21 /? or G 21 D, that is G is reduced or divisible.

Gαse 2.1. G is a divisible p-group. Then G ~ Σ,Ca where Ca is a p°° group.

If there is more than one summand, then there is a subgroup

Cί φ Oio, where G* is a proper subgroup of C α Q . Hence o(H) =o(G)9 but H is

not divisible, a contradiction. Therefore G is a p°° group in this case .

Case 2.2. G is a reduced p-group. Then G has a cyclic direct summand C

of order, say, pn. Zorn's lemma may be applied to s e t s S of cyclic groups

Ca of order pn such that Σ G α , Gα G S, exis ts and is pure in G (that i s , a

servant subgroup of G ) . There is then a maximal such set S*, and if X = C α ,

G α E S*, then X is a pure subgroup of bounded order. Hence K is a direct

summand, G - K + A. It is clear that /I has no cyclic direct summands of order

pn. This implies, by property ( P i ) , that o(A) < o(G)9 hence G 21 K. If, now,

n > 1, there is a subgroup H of K oί order o ( G ) such that H £ K. Therefore

596 w.R. SCOTT

Theorem 6 has a dual.

(P2 ) G is Abelian, and o(G/H) = o ( G ) implies G ~ G/H.

THEOREM 7. G has property {P2) if and only if ( i ) G is finite Abelian,

( i i ) G is infinite cyclic, ( i i i ) G is a direct sum of cyclic groups of order p,

( i v ) G is a p°° group, or ( v ) G is the direct sum of a non-denumerable number

of p groups.

Proof. If G is of one of the above five types, then it is clear that G has

property {P2).

Conversely suppose that G is infinite and has property ( P 2 ) .

Case 1. o(G/T) = o(G). Then, by (P2) G is torsion free. Let C be a cyclic

subgroup of G. Then 2C is cyclic, and G/2C has an element of order 2, hence

o(G/2C) < o{G). Therefore o(G) = K0,-and o(G/C) is finite, hence G is

cyclic.

Case 2. o (G/T) < o(G). Hence o ( T) = o ( G). Let S be a maximal linearly

independent set of elements, B the subgroup generated by S (set β = 0 if S is

empty). Then Γπ β = 0, hence Γ is isomorphic to a subgroup of G/B, and

therefore o(G/B) = o ( G ) . But G/β is periodic, hence G is periodic. It follows,

just as in the proof of Theorem 6, that G is either a divisible or a reduced

p-group.

Case 2.1. G is a divisible p-group. Then G = Σ C α , where C α is a p°° group.

If the number of summands is non-denumerable, we are done. If not, then G is

homomorphic to a p°° group, and o(G) = K0. Therefore by ( P 2 ) , G is a p°°

group.

Case 2.2. G is a reduced p-group. Then, almost exactly as in Case 2.2 of

Theorem 6, it follows that G is the direct sum of cyclic groups of order p.

REMARK. Szelpal [7] has shown that if G is an Abelian group which is

isomorphic to all proper quotient groups, then G is a cyclic group of order p or

a p°° group. Theorem 7 may be considered as a generalization of this theorem.

Szele and SzeΊpal [6] have shown that if G is an Abelian group such that

every non-zero endomorphism is onto, then G is a cyclic group of order p, a

p°° group, or the rationale. The following theorem may be considered as a

generalization.


( P 3 ) G i s A b e l i a n , a n d if σ i s a n e n d o m o r p h i s m o f G s u c h t h a t o(Gσ) = o ( G )

then Go — G.

THEOREM 8. G has property (P3) if and only if ( i ) G is finite Abelian,

( i i ) G is a p°° group, or ( i i i ) G is the group of rationals.

Proof. If G is of one of the above three types, then it is clear that (P3)

is satisfied.

Conversely, suppose that G is an infinite group satisfying ( P 3 ) .

Case 1. G is torsion-free. Then if pG ^ G for some p, the transformation

gσ~pg is an isomorphism of G into itself, so that o (Gσ ) = o (G), Gσ •£ G, a

contradiction. Hence pG - G for all p, and therefore G-ΣLRa9 where Ra is

is isomorphic to the group of rationals. If there is more than one summand, then

there is a projection σ of G onto ΣlRa9 CC ^ Cί0, a contradiction. Hence G is

the group of rationals.

Case 2. G is not torsion-free. Then G = A + B where A is finite (and non-

zero) or a p°° group. Thus the projection σ of G onto the larger of A and B yields

a contradiction unless B = 0. But in this case, since G is infinite, G - A is a

p°° group.

Finally (compare with Szele [5]) consider the following property.

(P4) G is Abelian, and if σ is an endomorphism of G such that o(Gσ ) = o(G)

then σ is an automorphism of G

COROLLARY. G has property ( P 4 ) if and only if ( i ) G is finite Abelian,

or ( i i ) G is the group of rationals.

REFERENCES

1. I. Kaplansky, Infinite Abelian groups, Michigan University Publications in Mathe-matics no. 2, Ann Arbor, 1954.

2. B.H. Neumann and Hanna Neumann, Zwei Klassen charakterischer Untergruppenund ihre Faktorgruppen, Math. Nachr. 4 (1950), 106-125.

3. W. R. Scott, Groups and cardinal numbers, Amer. J. Math. 74 (1952), 187-197.

4. , The number of subgroups of given index in non-denumerable Abeliangroups, Proc. Amer. Math. Soc, 5(1954), 19-22.

5. T. Szele, Die Abels chen Gruppen ohne eigentliche Endomorphismen, Acta. Univ.Szeged. Sect. Sci. Math. 13 (1949), 54-56.

6. T. Szele and I. Szelpal, Uber drei wichtige Gruppen, Acta. Univ. Szeged. Sect.Sci. Math. 13 (1950), 192-194.

598 w. R. SCOTT

7. I. Szelpal, Die Abe Is chen Gruppen ohne eigentliche Homomorphismen, Acta. Univ.Szeged. Sect. Sci Math. 13 (1949), 51-53.

UNIVERSITY OF KANSAS

ON HOMOGENEOUS LINEAR DIFFERENTIAL EQUATIONS WITH

ARBITRARY CONSTANT COEFFICIENTS

A. S E I D E N B E R G

Let K be an arbitrary ordinary differential field—for our purposes it is suf-

ficient to consider an arbitrary (algebraic ) field K which is converted into a

differential field by setting c ' = 0 for every c G K. Let u be a differential in-

determinate over K and let u — UQ9 uι, represent the successive derivatives

of u. Further, let Co9* 9cm be arbitrary constants over the field K\u)~

K(u0, uι9 •), that is, m + 1 further indeterminates with which we compute in

the usual way, setting cf = 0. In addition to the ring R ^ K{u \ ~ K [UQ$ uχ9 ],

we will also be interested in the rings Rt + m ~ & [UQ9 U\9 • , ι t£+ m ] . Theorems

referring to some one of these rings Rt+m may, if convenient, be regarded as

belonging to ordinary, rather than differential, algebra, but we will still apply

the operation of differentiation to elements of Rt+m (not involving ut+m). This

then amounts to a convenience in writing formulas.

Let IQ = CQ uo + + cmum. This element generates a prime differential

ideal [Zo 1 = ( h9 hi ) in S = K(c)\u\, where /; = c 0 U{ + + cm u; + m . We

a r e i n t e r e s t e d i n h a v i n g e x p l i c i t l y a b a s i s f o r [ l o ] n K \ u \ . I f Δ ( w ) i s t h e d e -

terminant of coefficients of any m + 1 of the Zt regarded as l inear forms in the

c;, then clearly A ( « ) G [ / 0 ] n K U ) and Theorem 2 below a s s e r t s that the

Δ ( u ) obtained from all choices of the Zj form the required bas i s .

Let us confine ourselves to the rings Rt+m

a n ^ $t +m = ^ ( c ) L uo? > ^ ί + m J

I n S ί + m , let p = (Z o , , lt)

LEMMA 1. p = ( l 0 , , lt) is an m-dimensional prime ideal in St + m .

Proof. Let G(UQ, , ut+m) G St+m. Eliminating success ive ly ut+m9

ut+m-W*9Um mod (l0, •• , Z ί ) , we may write G ( uθ9 , z^+ m ) Ξ G I ( M O > * * § >

um.\) mod (Zo» •••>/*), where Gi £St+m is a polynomial in the indicated vari-

ables . Moreover, starting with indeterminate values < . for uι, i = 0, ,wι —l,we

can build up a zero (ζQ, , ζt+m) of p by defining ^ m from the condition

Received December 7, 1953, This paper was written while the author was a Guggen-heim Fellow.

Pacific J. Math. 5(1955), 599-606

599

600 A. SEIDENBERG

/ 0 ( £ ) = 0 , and defining ζ + . successively from the condition Zί ( ^ r ) = 0 . Then

( ξQ, , ζt+m) is clearly a general point of p, whence p is prime and m-dimen-

sional.

LEMMA 2. Let p n Rt+m — P\ and let t >_ m — 1. Then P is a 2m-dimensional

prime ideal in Rt+m

Proof, Consider the equations:

+ + cm ξm = 0

From these we are going to solve successively for the c t , i — 0, , m — l

Since ξQ £ 0, we can solve for c0 and find CQ G K( c i , , cmt ζQ, , ζm)

Suppose in this way, solving success ively for the c t , we find

C O , , C J G K ( C J + I , . . . , c m , ξQ, •• , ί m + / ) , i < TO - 1 .

In fact, assume we have found inductively that

0 2 + 2 + K ( ^ 0 , . , ^ . + m ) cm

Since

dt X ( c 0 , " , c m , ^0,,•• , ^ m + ι . ) / X ( c 0 , > c m ) = m and

dt X ( c o , , c m ) / ί : = m + 1,

we have

dt K ( co, , cm$ ξQ, , ξmH )/K = 2m + 1

= dt

w h e r e d t s t a n d s for " d e g r e e of t r a n s c e n d e n c y " . F r o m t h i s w e s e e t h a t ξQ, •••,

ζm+i a r e a l g e b r a i c a l l y i n d e p e n d e n t o v e r K ( s i n c e t h e s e t c^ + l f , ί m + ι h a s

ON HOMOGENEOUS LINEAR DIFFERENTIAL EQUATIONS 601

2m + 1 members), in particular they are not zero. The coefficient of c + i in

li + i(ξ) is £2( + 1) plus a term in K( ξQ, • •, £ 2 ί + ι ) arising from c0 ξ^ι + +

since i + 1 < m, we have 2( ί + 1) < m + i + I and ^ ( ^ j ) ί

). Hence <?j+i G X( c t + 2, , fm+ + ι ) ; also î + i holds. Con-

tinuing, we have c 0 , , cm. L G K( c m , f0, , £ 2 m - ι )• Hence ίo» " » ^ 2 m . ι

are algebraically independent over K. Thus P is at least 2m-dimensional.

Let Δj ( ^ ) , i >_ m, be the determinant of the coefficients of the forms

IQ( ζ)J •» Zm. ι ( ^ ) , Zi(f) regarded as linear forms in c o , , c m ; that is,

V

Then one finds cyΔj (<f) = O, so that Δj(<f)=O. The coefficient of f + in

this equation is a polynomial in the indeterminates £Q9 , ζ this coef-

ficient contains the term ζQ ζ2 f2m_2

a n ( ^ n e n c e i s n o t z e r o (therefore also

ô( ζ}* * * ' ^m- l ( ί ) a r e linearly independent over X ( f ) ) . Thus P is at most

2m-dimensional, and hence exactly 2τn-dimensional, Q.E.D.

LEMMA 3. Let M = M(u) be the matrix:

α 0

, t >^rn.

Let /4 be the ideal generated in

olM(u). ΎhenA CP.

by the (m + 1) x (m + 1) subdeterminants

Proof. Since lo( ξ), , Zm. i ( £ ) are linearly independent over K(ξ) (and

in fact over any field containing K{ξ)) but Zo( ^ ) , , lm-ι ( ί \ 4 ' ( f ) a1"6

linearly dependent over K(ξ), the matrix M(^) has rank m. Hence A C_ P.

We want to prove A = P, in particular that /4 is prime. Conversely, if we

602 A. SEIDENBERG

knew that A were prime, we could conclude immediately that A - P. In fact,

suppose A is prime and let ηQ, , ηt+m be a general point of A. Since A has

a basis of forms of degree m + 1, no form of degree m vanishes at 77. Hence all

m x m subdeterminants of M(η) differ from zero, and it follows that A is 2m-

dimensional, whence A = P.

In proving A = P, we proceed by induction on m, the assertion being clearly

true for m = 0. For given m, we proceed by induction on t (£ >_ m). For ί = m,

we have to prove the following lemma.

LEMMA 4. Let D be the determinant

uί '

^m ' * ' U 2 m

Then D is different from zero and is irreducible in i?2m

Proof. By induction on m, being trivial for m — 0. D is linear in UQ, the

coefficient δ of UQ> being different from zero and irreducible by induction: in

particular, therefore, D 0. Also D is linear in U2m

a n ( ^ the coefficient δ ' of

U2m i s irreducible. D is reducible if and only if δ is a factor of D - woδ, hence

of D. Similarly for δ'. Now δ and δ ' are not associates, since they are of dif-

ferent degree in UQ SO D is reducible if and only if it is divisible by δδ'. For

m — 1, this means if and only if UQU2 —U^ is divisible by uoii2 This is not

the case. For m > 1, D is reducible only if it is of degree at least 2m, whereas

it is of degree m + 1. Hence for every m, D is irreducible.

DEFINITION. An ideal is called homogeneous if it has a basis of forms.

Similarly we call an ideal isobaric if it has a basis of isobaric polynomials.

LEMMA 5. A and P are homogeneous and isobaric.

Proof. A is clearly homogeneous. Moreover consider one of the (m + 1) x

(m + 1) subdeterminants of M{u)9 say one involving the ith. and th rows, i < j .

Then Mj+&_2 i s t n e element in the ith row and /rth-column and w/ + /.2 is the

element in the /th row and Zth column. Suppose k > L The determinant in ques-

tion has together with a term π wj+/c-2 "/+/-2 a l s o a t e r m ^π ' ui+l-2 ' M/+A>2»

which is of the same weight. Hence if rows i0, ,im are involved, each term

has the weight of the term uiQ z ^ + i ι2+2 ' ' wιm+m> ^ a t is> t n e determinant is


i s o b a r i c . T h u s A i s i s o b a r i c . As for P , we know that p i s homogeneous , and

from th i s and the fact that P = p n Rt+m one c o n c l u d e s immediate ly t h a t P a l s o

i s h o m o g e n e o u s . T o s e e t h a t P i s i s o b a r i c , l e t g(u)£P and write g{u) =

gr(u) +gr+ι(u) + •••, where gXu) i s zero or i s o b a r i c of weight /. It i s c lear ly

suff icient to prove gΓ(u) E Pf a s s u m i n g gr £ 0. Since g{u) E P , we have

h{c) g{u) ~ c, u) lί(c9 u),

where h(c) i s a polynomial in the c; a l o n e , and the A^ are polynomia l s in the

c; and uj We a s s i g n to c2- the weight m — i . L e t h{c) = hs(c) + Λ5 +1 ( c ) + •••,

where Ay(c) i s zero or i s o b a r i c of weight / and hs(c)^0. Observe that the

ll{cyu) are i s o b a r i c . Comparing terms of l ike weight on both s i d e s of the above

equat ion we s e e that hs(c) gr(u) = ^ ^ / ( c9u) l(( c9 u). H e n c e griu) G p.

THEOREM 1. A = P. In particular, therefore, for m > 0# A: uo ~ A.

Let

Proof. We proceed by induct ion on m and t, and first show that A:u0 -A.

be the general zero of P introduced above. L e t D(u) be the^.+

determinant occurring in Lemma 4. From D(ζ)=O we see that ζ2m

written as a quotient of two polynomials in the indeterminates ξQ,

with the denominator being

can be

' ^2m-l

d D 2 7 7 1 - 2

which is irreducible by Lemma 4. Hence we see that

ξ, ~LSn+l

(for were it zero, then ξ2m could be written as a quotient of two irreducible

polynomials in ζ^, , ζ , the denominator this time not being an associate

of the other denominator). Hence ξQ is algebraic over K( ζγy , ££+m) Hence

<î'# * ' ' < +m defines a 2m-dimensional prime ideal Pί in K[uϊ9 , ut+m]; and

Pi is generated by the (m + 1) x (m + 1) subdeterminants of M(u) which do

not involve the first row of M(u). Designating also by Pl9 the extension of

Pi to K [ UQ, , ut+m ], we see that Pγ C_ A. Let now ιiQg(u) G A. We write

604 A. SEIDENBERG

uog(u) = Σ / 4 J ( M ) ΔJ ( W ) , where the ΔJ ( M ) are the (m + l ) x ( τ n + l ) sub-

determinants of M(u), and the /4t are polynomials. We write Aι=A?+uoA."f

where A?does not involve UQ. We then have uo(g(u) - Σ>A?'Δi{u)) = Σ ^ . ' Δ j d ί ) .

The right hand side here i s of degree at most one in UQ, hence gχ = g(u) —

Σ,A?'Δι(u) does not involve u0: g v = g ι ( u i , , ut+m) Now g ( u ) and Δ ; ( u )

vanish at £ Q , . . , ξm+t, hence so does g t that is , g£ vanishes at ^ , , £ m + , .

Hence, g t G P t , whence g € A. Hence A: UQ - A.

As a corollary to the above we get that A : f = A for any polynomial

f £ Rm + t containing a term durQ, d £ K9 dÔ (m > 0 ) . For suppose fg £ A: to

prove g G /4. We may suppose / and g isobaric; and also homogeneous. We then

get duΓ

Q g G A$ whence g G /4.

We proceed to prove that /4 is prime. Let Zj = IΪ/UQ = c 0 v + * + cm Vj+m,

where v( = U(/UQ, We p a s s to the rings Rt+m ~ ^ [ ^ ι > * > vt+m J a n ( ^ ^t+m ~

K(c)[v] Observe that t> l y ,vt+m are algebraically independent over K.

Let if be the matrix of the coefficients of the Z, , that i s , the matrix:

V2

v2

V3

and let A be the ideal generated in Rt+m by the (m + l ) x ( m + l ) subdeter-

minants of M(v). Each such subdeterminant is a power of u0 times an (m + 1) x

(m + 1) subdeterminant of M(w); and vice-versa. It would therefore be sufficient

to prove A prime, in fact it would be sufficient to prove that the extension of

A to the quotient ring Q of /?t+ m relative to the ideal ( v Ϊ 9 , vt+m) i s prime.

For suppose this proved and g(u) h(u) G A, where we assume without loss of

generality that g ( w ) , h(u) are homogeneous. Dividing by appropriate powers of

UQ and sett ing

gU)/αJ=i(t;), h(u)/us

0=h(υ),

we get g(v)h(v) G A9 w h e n c e by a s s u m p t i o n f(v)g(v) or f(v)h(v), s a y

fg i s in A for s o m e f(v)ERt+m9 f £ {vi9 , vm). Mult ip ly ing by a power of

^0 w e find uζ f iu) g(u) G A, where f ( u) c o n t a i n s a term duζ. H e n c e g ( u ) G A.

T h e i d e a l /4 in /?£+ m h a s ζ^/ζ^ > ^t+rdô a s a z e r o » n e n c e i s a t l e a s t

( 2 m — 1 ) - d i m e n s i o n a l . A l so A r e m a i n s at l e a s t ( 2 m — 1 ) -d imens iona l upon ex-

t e n s i o n to Q. In fact , if ξχ/ξ0, •••, ζf^m/ô d e t e r m i n e s P in Rt+m, t h e n


P C_{v\ 9 9vt+m), as one s e e s from the fact that £ 0 , » , £ ί + determines a

homogeneous and i sobar ic ideal P and UQ $L P.

Subtracting v{ t imes the first row from the (ί + l ) t h row of M, we get the

matrix

v i v 2

0

Each (TO + 1) x (TO + 1) subdeterminant of this matrix is also an (TO + 1) x

(m . + 1) subdeterminant of /I/. Hence one sees that every m x m subdeterminant

of the matrix

v2

vt+m

is a leading-form of an element in Q A, These m x m subdeterminants generate,

by induction, a 2(m — 1 Vdimensional prime ideal in K\_v2j »vt + m\ a n <^

hence a (2m - 1 )-dimensional prime ideal q in K[vι9 , Vί + m l . The leading

form ideal of A contains or equals ~q. If it contained ~q properly, it would be of

dimension less than 2m — 1. But an ideal and its leading form ideal have the

same dimension [1; Satz 8], Hence q is the leading-form ideal of A and A is

{2m - l)-dimensional.

Moreover A is prime. For quite generally in a local ring, if an ideal A has a

prime ideal ~q as leading form ideal, it must itself be prime. In fact, suppose

gh E A$ g jέ A% h<t A. Then the leading form ideal LFI(A9g) of (A9g) contains

^properly, and likewise for (A,h). But LF1 (A, g) x LF1 (A, h) C LFI ( ( J , g )x

(A$h)) C_ LFIA = q, a contradiction. Hence /4 is prime, and the proof is com-

plete.

The following theorem is an immediate consequence of Theorem 1.

T H E O R E M 2. A basis for ilo]n K\u\ is given by the {m + 1 ) x (m + 1 )

subdeterminants of the oo x (m + 1 ) matrix

um

u2

606 A. SEIDENBERG

REFERENCE

1. W. Krull, Dimensionstheorie in Stellenringen, J. Reine angew. Math. 179 (1938),204-226.

UNIVERSITY OF CALIFORNIA!

BERKELEY, CALIFORNIA

CANTOR-TYPE UNIQUENESS OF MULTIPLE TRIGONOMETRIC INTEGRALS

VICTOR L. SHAPIRO

1. Introduction. It is the purpose of this paper to obtain results in Cantor-

type uniqueness for multiple trigonometric integrals similar to those obtained

previously for multiple trigonometric series ([5, 11, 12]). As might be expected,

the results in the integral case are a bit more difficult to obtain.

Vectorial notation is used for the most part throughout this paper. Thus u

designates the point in ^-dimensional euclidean space, En, with coordinates

( u ί 9 9 u n ) , t h e s c a l a r p r o d u c t ( u 9 x ) = u± x ι + + un x n , w i t h \u\={u9u)^2

and u + 0.x is the point (u ι + axί9 , un + CLxn ).

Previously the author [13], using equisummability between trigonometric

integrals and trigonometric series, has obtained in the special case of double

trigonometric integrals the following result:

Let c{u)9 in L2 on any bounded domain, be 0 ( | u | ), e > 0. Suppose the

double trigonometric integral /„ e ' u c{u) du is circularly summable (C9 1)

to f(x). Furthermore suppose fix) is in Lip Cί, (X > 0, on every bounded

domain id depending on the domain). Then the double trigonometric integral

e-ί{x>u)fix)dx

is spherically summable ( C , 1) to c iu) for almost every u.

Specializing fix) to be the zero function (which is what is meant by Cantor-

type uniqueness, [ 15, p. 274]) and using a more direct attack on the problem,

we are able in this paper both to weaken the hypotheses of the above theorem

as well as to extend the results to ^-dimensional integrals.

2. Definitions and notation. The open π-dimensional sphere with center x

and radius r will be designated by Dnix,r), and the surface of the sphere by

Cn(x$r).

Received November 26, 1953. Presented to the American Mathematical SocietyNovember 27, 1953. This investigation was supported in part by a grant from the RutgersUniversity Research Fund.

Pacific J. Math. 5 (1955), 607-622

607

608 VICTOR L. SHAPIRO

Following Bochner [ l ] , we shall say that the multiple trigonometric integral

fE c{u)eι XfU du is spherically convergent at the point x to the finite value

Lix) if the spherical partial integrals of rank R converge to L(x), that is if

(D /*(*)= ί ei{x>u)cU)du—>L{x) (asR—>oo.)JDn(0,R)

The integral

( 2 ) < 4 α ) U ) = 2 α β - 2 α ίR lr(x){R2-r2)a-ιrdr, α > 0 ,κ Jo

is called the (C, cc)-mean of rank R of the multiple trigonometric integral

fE c(u)e 9 du$ and this integral is said to be spherically summable (C, α)

t o L U ) i f σ < α ) ( z ) — > L ( « ) as/?—>oo.

Given F{x) integrable on Dn{xθ9r), we designate the mean value of F in

this sphere by A (F; xo;r). Given F(x) integrable on CU(XQ; Γ), we designate

the mean value of F on this surface by L{F; xo; t). Thus, designating the

volume of the unit ^-dimensional sphere, 2 πn /nV(n/2), by Ωn and the (n — 1)-

dimensional volume of its surface, 2πn /Γ(n/2), by ωn, we have

A(F;xo;r) = (Qnrn)'1 I F(x)dx

JDn(x0,r)(3)

L(F;xo;r) = ω"1 / F(x0 + rx)dSn. t(x)

where dS^ί is the (n — 1) dimensional volume element of C n (0,1) .

We set

V l ( F ; * o ; r ) = L ( F ; * o ; r - ) - F ( * o ) and V2 ( F;x0; r) = A ( F; xo; r) - F(x0 )

and say that F(x) has a generalized Laplacian of the first or second kind at

the point x0 equal to OCi or 0,2, respectively, if

lim 2n Vι{F;x0;r)/r2 = C^

or

lim 2(n + 2) V2 ( F ; xo; r)/r 2 = a2

CANTOR-TYPE UNIQUENESS OF MULTIPLE TRIGONOMETRIC INTEGRALS 6 0 9

The generalized Laplacian of the first and second kind of F at x0 will be

designated by ΔιF(x0) and Δ 2 F ( % 0 ) , respectively. It is known, [6, p. 261],

that if Fix) is in class C ( 2 ) on Dnix0, r0 ), then ΔF(% 0 ) = ΔιFix0 ) = Δ2F ix0 )

where ΔF(%) is the ordinary Laplacian of F at x.

The closure of the set W is designated by W; and its characteristic function

by \ψix) The set Z is said to be a closed set of vanishing capacity if for

every r§ ZDw(0, r) is a closed set of capacity zero. It is known, [4] , that if

Z is a set of vanishing capacity then Dn(xθ9 r) - ZDniχθ9 r) is a domain.

The trigonometric integral fE e fU c iu)du is said to be of type (U) on

a domain G if

/ ei(x'u)c(u)\u\-2duJEn-Dn(0,\)

converges spherically on G to a function Fix) which is continuous on G.

Throughout this paper En stands for n-dimensional euclidean space where

n > 2, and μ = in - 2)/2.

The function 7j(r) is the Bessel function of the first kind of order i.

3. Statement of main results. We shall prove the following two theorems

concerning Cantor-type uniqueness for multiple trigonometric integrals.

THEOREM 1. Given the multiple trigonometric integral L e fU c iu) du

where ciu) is a complex-valued function which is integrable on every bounded

domain. Let Z be a closed set of vanishing capacity. Suppose that

( i ) The integral is spherically summable ( C, 1) to zero almost everywhere.

( i i ) The (C, 1) spherical mean of rank R, σ^Hx), is such that l i m ^ ^ ^

| ^ l ) U ) | < oo inEn -Z.

( i i i ) c iu) i I u \2 + 1 ) " 1 is in L^ on En.

Then ciu) vanishes almost everywhere.

THEOREM 2. Given the multiple trigonometric integral JE eτ *u ciu)du

where ciu) is a complex-valued function which is integrable on every bounded

domain. Let Z be a closed set of vanishing capacity. Suppose that

( i ) and ( i i ) The same as ( i ) and ( i i ) of Theorem 1.

( i i i ) The integral is of type iU) on En.

( i v ) ciu)i\u\2 + I)"1 is in L2 on En.

Then ciu) vanishes almost everywhere.


For the special case of the plane, we prove the following theorem.

THEOREM 3. Given the double trigonometric integral fE e fU c(u)du

where c{u) is a complex-valued function which is integrable on every bounded

domain. Let Z be a closed set of vanishing capacity and W be a closed de-

numerable set such that WZ = 0. Suppose that

( i ) The integral is spherically summable ( C9 1) to zero in E2 — Z,

( i i ) The integral is of type (U) on E 2 — W.

( i i i ) c ( u ) = o ( I u I ) as \u\—» oo

( i v ) c ( u) ( I u I 2 + I ) " 1 is in L2 on En.

Then c(u) vanishes almost everywhere.

4. Fundamental lemmas. Before proving the main theorems of this paper, it

is first necessary to establish a connection between the ( C, 1) spherical sum-

mability of the integral fE e ' u c (u) du and the generalized Laplacians of

the "anti-Laplacian" of this integral. In short, we shall now establish some

lliemann-type, [15, p. 270], results for the multiple trigonometric integrals.

We need prove the following lemma only for the plane, since the conclusion

is hypothesized for Theorems 1 and 2.

LEMMA 1. Let du) be a complex-valued function which is integrable on

every bounded domain in the plane, vanishes in D2{0,ro), r 0 > 0, and is o\\u\)

Suppose that σ ^ ι ) {x0 ) = o {R) where σ^ ι ) (:x;) is the ( C , l ) spherical mean of

rank R of fE e

i(x>u)c(u) du. Then fE ei(x>u) c U ) \u\'2 du is spherically

convergent.

Without loss of generality, we assume x0 to be the origin. Then with //?(%)

given by (1) and σ^Hx) by (2), we have

= 2 ίR r-3lr(0)dr+R-2IR(0)Jo


Since by assumption σ^ ι ) ( 0 ) = o ( R ) , to prove the lemma it only remains to

show t h a t / Λ ( 0 ) = o ( β 2 ) . But

IR(0) = [(/? + I ) 2 σ

- ( 2 R + I ) ' 1 f c ( u ) [ R + D 2

JD2 (θ,R + l ) -D 2 (θ,R)

and the proof is complete.

LEMMA 2. Let c{u) be a complex-valued function which is integrable on

every bounded domain in En and which vanishes in Z ) π ( 0 , ΓQ ), ΓQ > 0. Suppose

that

( i ) lίm D ^ I On (XQ ) I - d where σ^ (XQ) is the (C9 1 ) spherical mean of

rank R of fF e 'u* c (u)du and d is finite-valued.

( i i ) - f ei{x'u)c(u)\u\-2du

is spherically uniformly convergent in Un(xOf ί0 ), ί0 > 0, to F (x).

Then l i n i ί ^ o I %n^ι (F; xo; t )/t2 \ <_Kd where K is a constant independent of

%o and d.

Observing that for fixed u

(see [1, p. 177]), we have by assumption (ii) for t sufficiently small that

L(F;xo;t)

= -2^T(μ+l) lim / eiix°>u)c(u)\u\-2jΛ\u\t){\u\t)'μduRôoJDn(o,R)

and consequently that

(4) (2n)Vι(F;x0;t)/t2= l im / e i { x Q ' u ) c(u)η{\u\t)duRôoJDn(o,R)

where

( r ) r - μ ] / r 2 f o r r > 0, τ ; ( 0 ) = l ,


and η(r) i s in C .

Making the following observations:

( a ) By the second mean-value theorem applied to the real and imaginary

parts of IR(XQ) given by ( 1 ) and hypothesis ( i i ) we have IR(XO) = O(R2)9

( b ) F o r f i x e d t9 η(Rt)=O(K'2) a n d η'(Rt) = O(R'5/2) w h e r e τ j ' ( r ) =

dη(r)/dr, we o b t a i n from ( 4 ) t h a t

( 5 ) 2nyι(F;x0;t)/t2 = r ί [°° r2 σ < ι > ( * 0 )t3 Oi(rt) drJ Γo Γ

where OC (r) = dr"177 ' ( r)/dr.

From the fact that Cί(z) is an entire function of the form Σ = 0 b(Z21 l, we

have that there exists a constant Kx such that

(6) | ( χ ( r ) | < «!Γ for r < 1

From the fact that Jμ(r) =0{r'i/2) as r —> 00, and

rfr-μ/μ(r)/rfr = - r " μ / μ + 1 ( r ) f

we obtain that there exists a constant K2 such that

(7) | α ( r ) | < K 2 [ r - ( μ + 7 / 2 ) + r- S ] f o r r > l

From (5), (6) , and (7) , the conclusion of the lemma follows readily. For

given an e > 0, choose RQ so large that \o^Hxo) \ <_ d + 6 for R > Ro. Then

for t < RQ1

9 it is easily seen that

(8) \ 2 r f t ι ( F ; x O t t ) / t 2 \ < K ( d + e ) + O ( ί 4 )

where K is a constant independent of xOi d9 and e. Taking the limit superior

of the left side of ( 8 ) as t —» 0 and then the limit of the right side as e —> 0,

we have the proof of the lemma.

LEMMA 3. Let the hypotheses be the same as in Lemma 2 except that

For if d = 0, the lemma follows immediately from Lemma 2. If d £ 0, choose

cι(u) integrable on En, vanishing for u in [En - Dn(0, 2 ) ] + Dn(0,1), and such

that fE Cι(u)ei{x°'u) = d. Set F ι ( * ) = - / £ Ci(u) \u\'2 ei(x'u)du. Then


0 = AÂxo)- /iίF^xo) = Δ ^ U o ) - Δ F ^ Λ Q) = ΔιF(xo)-d.

LEMMA 4. Let c(u) be a complex-valued function which is integrable on

every bounded domain in En and which vanishes in Dn(0fro)9 ro> 0. Suppose that

( i ) σ^Hx), the (C, 1) spherical mean of rank R of JP eι XfU c (u)du,

is such that lim/? _» oo | σ^ (Λ O ) I = d

( i i ) c (u ) I u I" is in L2 on En.

( i i i ) - f eί(x°>u)\u\-2c(u)duJ

is spherically convergent to F(XQ). Set

i(x'u)c{u)\u\-2du.fDn(0,R)

Then

ϊhn | 2 ( 7 i + l ) V 2 ( F ; « 0 ; i ) A 2 | < Kd

where K is a constant independent of XQ and d.

Setting

TR(x)=- f ei(x>u)c(u)\u\-2du,JDn{θ,R)

observing that A(F;xo;t) - l i m R _ o o A ( TR xo; t) and that for fixed u,

i { \ nntn Γ rn-ιL(ei{x'u);x0;r)dr

Jo

we obtain

( 9 ) A(F;xo;t)

= - lim (μ)f

JDn(0,R)

and consequently

6 1 4 VICTOR L. SHAPIRO

lίm f ei{x°>u)γ(\u\t)c(u)duRôcJDn(0,R)

here

U + l ) ( ^ l ) r ) ] / r 2 for r > 0, y(0) = l ,

and γ(r) i s in C ( o ° I

S i n c e γ{r) h a s t h e s a m e form a s η(r) in L e m m a 2 with μ r e p l a c e d by μ + 1,

we c a n p r o c e e d a s in t h a t l emma a n d o b t a i n

2 U + 2 ) V 2 ( F ; * 0 ; ί ) / ί 2 = 2 - 1 [°° t3r2 σ ( 1 ) (x0) β(rt) drJo

where β(r) = dr" γ'(r)/dr. Then we can proceed in a similar manner to obtain

that for e > 0

ϊ ϊ m " \2(n + 2)\72(F;x0;t)/t2\ <K(d + e)

where K is a constant independent of λ'o, d, and e. Since e is arbitrary the con-

clusion of the lemma follows.

LEMMA 5. Let the hypotheses be the same as in Lemma 4 except that

lim/ôo O^HXQ ) = d. Then Δ 2 F(% 0 ) = d.

In the same manner that we obtained Lemma 3 from Lemma 2, we obtain

Lemma 5 from Lemma 4.

LEMMA 6. Let F{x) be real-valued and continuous on Dn(xo,ro ), r0 > 0.

Suppose that

( i ) A 2F(%) = 0 almost everywhere in L)n(x0$r0 )

( i i ) Tϊm \2(n + 2) V2 (F; x; r)/r2 \ < oo for all x in Dn(xθ9ro).r —* oo

77ιeπ F ( # ) is harmonic in Dn(xOiro ).

Following the pattern of proof in [9], we give a proof for n >_ 3.

To prove the lemma, it is sufficient to show that Fix) is subharmonic in

Set

CANTOR-TYPE UNIQUENESS OF MULTIPLE TRIGONOMETRIC INTEGRALS 615

f(x)=2(n + 2)[ ϊ h n ~ V 2 ( F ; % ; r ) / r 2 + lim V2 ( F; x; r)/r2 1/2Γ->0 Γ-»0

for x in 0n(χ0i r0 ). Then / (A; ) = 0 almost everywhere in Dn(x0; r0 ).

By the theorem of Vitali-Caratheodory [10, p. 75] , there exis ts a sequence

of nondecreasing upper semίcontinuous functions ίgm(χ)\ such that gm(x) <

f (x) for a l l x in Dn{xo,ro), gm^x^—> f (x) a l m o s t e v e r y w h e r e in Dn(xOfro),

g ( A : ) i s i n t e g r a b l e on Dn(xoiro)9 and s u c h t h a t

limm

im / £ ( x ) dx — I f (x) dx for r < r0 .-oo JDn(x0>r)*m JDn(xQ,r)

Set

^gJx)=-[ωnU-2)Yιf gmU)\u-x\2-ndu.

Then Δ~ιgm(x) is superharmonic, since gm(u) £ 0 for almost all u in Dn(xQ$ ro)

Furthermore, we observe that for fixed u

A{\x-u \2'n;x0;r) = | x 0 - u \ 2n if |%0 - M | > r

= / ι r " Λ 2 - 1 [ r 2 + |%0 - α | 2 ( 2 - Λ ) τ ι - 1 ] if \x0 ~u\ <r.

Consequently, for xι in Dn(xo,r) with r sufficiently small,

( 1 1 ) V 2 ( Δ - 1 g m ; x ι , r ) = [ ω R ( r a - 2 ) ] - 1 ^ ^ ^gju) \ \u - X ι \2'n

- r a r - " 2 - 1 [ r 2 + |SB t - u | 2 ( 2 - n) n ι ] }rfu.

Suppose g_,(*i ) s finite. Then by the upper semi-continuity of gm(u) at

%i, for e > 0 and r sufficiently small, we have from ( 1 1 ) that

V 2 ( Δ - ι g m ; * i ; r ) < [ g m ( X ι ) + ε ] [ ω n ( n - 2 ) ] - ι [ ω n ( n - 2 ) ] r 2 / 2 { n + 2 ) .

Consequently, we conclude that

( 1 2 ) lim 2 U + 2 ) V 2 ( Δ " l £ xι; r)/r2 < g(xι).Γ->0

S i m i l a r l y , in c a s e g (%i ) = — oo, c o n c l u s i o n ( 1 2 ) r e m a i n s v a l i d .


From the fact that Δ" g ix) is superharmonic, we have that F — Δ" g is

upper semi-continuous in Dnix0, r 0 ) . From ( 1 2 ) we conclude that

Tϊm" 2 U + 2 ) V 2 ( F ~ Δ - l g ; % ; r ) / r 2 > 0 for x in D Λ ( * o , r o ) .Γ-»0

Therefore by [8, p. 14], ί F — Δ" g 1 is a nondecreasing sequence of sub-

harmonic functions in Dnixθ9 ro ). But limm_>(X) Δ" g ix) = 0 almost everywhere.

Therefore Fix) is almost everywhere equal to a subharmonic function, Gix), by

[8, p. 22]. But A(F x r) = AiG x r) —> Gix) for all % in Dnixθ9 r0 ). However

from the continuity of F we have A(F x r) —» F(%), and the proof of the lemma

is complete for n >_ 3. For β = 2 a similar proof can be given with the Newtonian

potential replaced by the logarithmic potential.

For the case of the generalized Laplacian of the first kind, we have a similar

lemma with a similar proof, see [9],

LEMMA 7. Let Fix) be real-valued and continuous on Dnixθ9 r0 ), r0 > 0.

Suppose that

( i ) Δ xF ix ) = 0 almost everywhere in Dnixθ9 r0 ).

( i i ) lim \2n^χ (F; x; r)/r2 \ < oo for all x in Dnix0; ro ).

Then Fix) is harmonic in Dnixθ9 ro )•

We now prove s o m e l e m m a s c o n c e r n i n g t h e s p h e r i c a l s u m m a b i l i t y iC9n) of

F o u r i e r t r a n s f o r m s .

LEMMA 8. Let Gix) be a function in L\ on En which vanishes in Dni0,ro),

r0 > 0. Suppose that Fix) = fE eι * Giu)du is in C ( 2 ) on En. Then for u in

Dnio9r0/2)-0

(13) j [e'i(x>u)Fix)~i-e i(x>u)\u\'2AFix))]dx

is spherically summable iC9n) to zero.

For, by Green's second identity, we have

(14) » ( « ) - f [e i(x>u) F (x) - {-e-i(x>u) \u\ 2 AF(x))]dxJD(0R)f

JDn(0,R)

"-1 f F{Rx)i(x,u)e-iR{x'ιι)dSn.ι(x)/cπ(o,i)


+ R n ' 1 f dF(Rx)/dR e'iR(x'u)dSnml(x)\ Ξ \u\'2(AR +BR)JCn(θ,l) i

where dSn_ι(χ) is the (n — 1) dimensional volume element on the unit sphere

CB(O,1).

We shall now show that both AR and BR are ( C, n) summable to zero. For,

by Fubini's theorem, we have

( 1 5 ) {MR2Yι

lifJεn-Dn{o.ro)

=(MR2)

where M =(2π)n/2/2n ι (n-1)1 and

1 for 0 < r < 1

φι(r) =0 for r > 1

( 1 - r 2 ) " - 1 forO < r < 1

0 for r > 1

Since for fixed u £ 0, (x, u) is a homogeneous polynomial which is also a

harmonic function in x, we have by [2, p. 806] and [14, p. 3731 that the right

side of (15) is equal to

(16) G(γ)( y - u9u) \y ~ u \)dy

\y ~u ( Λ l y - u )"

Clearly ( 1 6 ) tends to zero a s R —»oo; so AR is (C,n) summable to zero

for u in Dn(0, r0 / 2 ) - 0.

We also observe after integrating by parts that

17) (MR2)'1 [R rψl-)BrdrJo n\Rl

From the above discussion concerning AR and from [ l , Theorem 1], to show

that BR is (C, n) summable to zero for u in Dn{0,r0/2) - 0 , it is sufficient to

show that


(18) (MR2)'1 f F(x)φ ( i l l ) ! l ! _ «>-'•<*.«>,&—»0 as /?—• ooJπ " - 1 \ P / r>2

But by [2, p. 806] and [14, p. 373] the expression in (18) is equal to

En.Dn(θ.ro) l y - u l " " 1 n-μ<- I(R\y-u\)

- K2 dy.

(R\y-u\n-μ-2) \

where Kt and K2 are two constants depending on n.

Clearly (19) tends to zero as R—>oo for u in Dn(0, r o /2) - 0; so BR is

(C, n) summable to zero and the lemma is proved.

LEMMA 9. Let G{x) be a function in L2 on En which vanishes in ^n(0, r υ ) ,

r 0 > 0. Suppose that fE eι *u G(x)dx is spherically convergent to a function

F(χ) which is in C ( 2 ) on En. Then for u in Dn(0,r0/2) - 0

f

is spherically summable (Cfn) to zero.

For (14) also holds in this case, and as in Lemma 8, we have to show that

both AR and BR are (C, n) summable to zero.

Since both F(x) and φ (\x \/R)(x$u) are in L 2 on En, ParsevaΓs formula

gives us both (15) and (16). We therefore conclude as before that AR is ( C, n)

summable to zero for u in Dn(0, ΓQ/2) - 0.

To show that BR is summable (C9n) to zero, we obtain (17) as in Lemma 8.

Then from the fact that AR is ( C, n) summable to zero and from [3, Theorem

55], it is sufficient once again to show that (18) holds.

But by Parseval's formula, we obtain that the expression in (18) is equal

to (19). Observing that for u in Dn{Q, ro/2) - 0 and for y in En -Dn(0,r0) there

exists a constant Kn such that

I W - i ( Λ | y - « P I <κn(R\γ-u\Ti/2 for/? > l

a n d t h a t for s u c h u9 \y -u\2~n i s in L 2 on En - Dn(Q,r0), we c o n c l u d e t h a t


(18) holds and consequently that BR is ( C9 n) summable to zero, which proves

the lemma.

5. Proof of Theorem 1. To prove Theorem 1, it is sufficient to show that

for any r0 > 0, c(u) = 0 almost everywhere in Dn(0y r o /2) . Set

*( \ f [ « ' ( * ' B ) - i - * U»>1 , u

Fι{x) = -I c(u)duJDn(0,rQ) | u | 2

Then, Fχ(x) is in C °° ' on En and

AF t(%)= / eilx'u)c(u)du.JDn(o,ro)

Set

./ \ C ( u )

which is by ( i i i ) continuous in En. Then by Lemma 2 and ( i i ) ,

Tίm | 2 n V ι ( F 2 ; % ; r ) / r 2 | < oo

r-»o

in En - Z and by Lemma 3 and ( i ) , Δ 1 F 2 ( Λ ; ) = - Δ F 1 ( Λ ; ) almost everywhere.

Set F ( Λ ) = Fχ(x) + F2 (x). Take any χ0 in F n and consider Dn{xQir\)>

ri > 0. From the definition of a closed set of vanishing capacity, we see that

there is a closed bounded set of capacity zero Zγ such that

lim \2rNι(F;x;r)/r2\ ^{AF^x)]* lim \2rtfi ( F 2 ; x; r)/r2 \ <

for x in the domain G = Dn(xθ9 ry) — ZιDn(χ9n) Furthermore almost everywhere

in Gf ΔχF(x) = ΔF t (%) + ΔiF 2 (%) = 0. Consequently it follows from Lemma 7,

that F(x) is harmonic in the domain G = Dn(x<)9 r\ ) — ZιDn(xθ9 r ). But F(x)

is continuous in Dn(x 0>ri)« Therefore by [7, p.335], F(%) is harmonic in

Dfji Oi Γ ι ) a n ( i since xQ is arbitrary, F ( Λ ) is harmonic in En

From the fact that F{x) is harmonic in En, we now have that F2(x) =F{χ)

-Fx(x) is in C ( o o ) on F w and that ΔF2 (%) = - ΔFX (%) for all x. Also by [ l ,

Theorem l ] we obtain that


[2πYn f e-i(x u)F2(x)dx

is spherically summable (C,n) to zero for u in Dn(0, ro/2) — 0. Therefore by

Lemma 8 for such u,

f e-i{x u)[-ΔF2(x)]dx

is spherically summable (C, n) to zero. But for almost all such u9 we have that

(2πYn f e-i(x'u)AFί(x)dx

i s s p h e r i c a l l y summable ( C , n) to c{u) S ince Δ F ^ Λ ; ) = — Δ F 2 ( % ) , we c o n c l u d e

t h a t for a l m o s t a l l u in Dn(0, ΓQ/2), C(U) = 0, which p r o v e s the theorem.

6. Proof of Theorem 2. T h e proof i s quite s i m i l a r to t h a t of T h e o r e m 1.

O n c e a g a i n it i s su f f ic ient to prove t h a t for any r 0 > 0, c(u) -0 a l m o s t every-

where in Dn(0, ro/2).

Set

F ι ( % ) = - / [ e i ( * tt)-l-ί(*,α)] — du,

and

F 2 ( % ) = - lim ί ei(x>u) ίί^ldu.R-.ocJDn(o,rR)-Dn(o,r0) | α | 2

By (ii i), F2(x) is continuous. Then in a manner exactly analogous to the proof

of Theorem 1 except that Lemmas 4, 5, and 6 are used instead of 2, 3, and 7,

we obtain that F2(x) is in C^°°^ and that ΔF2 {x) = - Δf\ (x). By Lemma 9

and [3, Theorem 55], we obtain that / £ e " ι ^ ' w ^ [ ~ Δ F 2 (x) ]dx is spherically

summable (C$n) to zero for u in Dn(0, ro/2) - 0 . But by [1 , Theorem 1] for

almost all such u, we have that

(2 πyn ί

is spherically summable ( C , n) to C ( M ) . Since ~ Δ F 2 (A;) = Δ F t (x), we con-

clude that c ( u ) = 0 almost everywhere in Dn(0,ΓQ/2) and the theorem is proved.


7. Proof of Theorem 3. L e t F{ix) be a s in Theorem 2 with n r e p l a c e d by

2, and le t

F 2 U) = - U m /R-oo JD2{θfR)-D2(θ,ro) \u\Z

where r0 > 0. This limit exists for x in Z by ( i i) and for x not in Z by ( i ) ,

(i i i), and Lemma 1. Furthermore by (i i) F2ix) is assumed continuous in E2 - W.

It is clear from the proof of Theorem 2 that to prove this theorem we need only

show that F2ix) is continuous in E2 or what is the same thing that Fix) =

Fγ ix) + F2ix) is continuous in E2.

By (i i) Fix) is continuous in E2 -IF, and by Lemmas 5, Δ2Fix) = 0 in

E2 - Z. Let D2ixOfrι ) be any disc which has a null intersection with W. Then

as in the proof to Theorem 1, Fix) is harmonic in this disc and consequently

in E2 - W. We also observe that now Δ 2 F(%) = 0 in the whole plane and further-

more that Fix) is in L2 on any bounded domain.

Let Wι be the set of discontinuity points of Fix) and let XQ be an isolated

point of Wι» Then there is a closed disc D2ixo,r2) whose intersection with

Wι is xQ. Then by the above discussion we have that Fix) is in L2 on D2ix0% r 2 ),

harmonic in D2ix0i r2) - XQ, and satisfies the further condition that Δ 2 F(% 0 ) = 0.

Consequently by [12, Lemma 4], Fix) is then harmonic in the whole disc and,

a fortiori, continuous at XQ.

Therefore Wl9 has no isolated points and Wι is a perfect set. But W\ C W is

at most denumerable, and by [10, p. 55], Wx is then the empty set. Thus Fix)

is continuous in the whole plane, and, as mentioned above, the proof of this

theorem is reduced to that of Theorem 2.

8. Appendix. In closing we point out that the assumption W and Z have a

null intersection in Theorem 3 is a necessary one. For consider the double

trigonometric integral fE c iu) eι^x'u'du with C ( M ) = 1 . ( i i i) and (iv) of

Theorem 3 are clearly satisfied. Observing that the spherical mean of rank R,

(%), w i th Λ; 5 0 is given by

£ ) ~4πJ2i\x\R)\x\-2=OiR'ι/2),

we see that ( i ) is satisfied with Z equal to the origin. Furthermore, we observe

that for x £ 0

lim / \u\-2ei{x>u)du = 2πfO°i Joir)r'ιdr.R-+OG JD2(O,R)-D2iθ,l) J\χ\


Consequently ( i i ) is satisfied with W consisting of the origin. But W and Z do

not have a null intersection, and the conclusion of Theorem 3 does not hold.

REFERENCES

1. S. Bochner, Summation of multiple Fourier series by spherical means, Trans.Amer. Math. Soc. 40 (1936), 175-207.

2. , Theta relations with spherical harmonics, Proc. Nat. Acad. Sci., 37(1951), 804-808.

3. S. Bochner and K. Chandrasekharan, Fourier transforms, Princeton, 1949.

4. M. Brelot, Sur la structure des ensembles de capacite nulle, C. R. Acad. Sci.Paris 192 (1931), 206-208.

5. M. T. Cheng, Uniqueness of multiple trigonometric series, Annals of Math. 52(1950), 403-416.

6. R. Courant and D. Hubert, Methoden der mathematischen Physik, vol. 2 Berlin,1937.

7. O. D. Kellogg, Foundation of potential theory, Berlin, 1929.

8. T. Rado, Subharmonic functions, Ergebnisse der Mathematik, vol. 5, no. 1,Berlin, 1937.

9. W. Rudin, Integral representations of continuous functions, Trans. Amer. Math.Soc. 68 (1950), 278-286.

10. S. Saks, Theory of the integral, 2d. ed., Warsaw, 1937.

11. V. L. Shapiro, An extension of results in the uniqueness theory of double trigo-nometric series, Duke Math. J. 20 (1953), 359-366.

12. , A note on the uniqueness of double trigonometric series, Proc. Amer.Math. Soc. 4 (1953), 692-695.

13. M Summability and uniqueness of double trigonometric integrals, Trans.Amer. Math. Soc, 77(1954), 322-339.

14. G. N. Watson, A treatise on the theory of Bessel functions, Cambridge, 1944.

15. A. Zygmund, Trigonometrical series, Warsaw, 1935.

RUTGERS UNIVERSITY, NEW BRUNSWICK, NEW JERSEY AND

THE INSTITUTE FOR ADVANCED STUDY, PRINCETON, NEW JERSEY

MINIMAL BASIS AND INESSENTIAL DISCRIMINANT

DIVISORS FOR A CUBIC FIELD

L E O N A R D T O R N H E I M

In terms of the coefficients OC, jS, γ of a defining equation

of a cubic field F over the rational number field Q9 Albert [ l ] has given an ex-

plicit formula for a minimal basis, that is, a basis of the integers of Q{θ)

over the rational integers. We solve this same problem with a shorter proof and

a simpler result. This basis is then used to find the maximal inessential dis-

criminant divisor, that is, the square root of the quotient of the g.c.d. of the

discriminants of all integers of Q(θ) by the discriminant of Q(θ). It is known

[3] that the only prime dividing it is 2; we determine the power as 2° or 2 ι .

We first secure a normalized generating quantity,

L E M M A 1. If K is any cubic field, then K = Q(θ) with

( 2 ) 6>3 + aθ2 + 6 = 0 ,

where ( i ) a and b are rational integers, ( i i ) no factor of a has its cube dividing

b9 and ( i i i ) if 3 \\a, then the discriminant Δ = - b ( 4 α 3 + 27 b) of θ is not di-

visible by 3 4 unless 3 | b.

Here gn |1 y means gn | γ and gn l \γ.

Proof. The substitution θ'~ θ+ α/3 is used to obtain an equation of form

(1) with α zero. Follow this by the substitution 0 '= 1/0 to obtain (2) . For

Conditions ( i ) and ( i i) it is obvious that a substitution 0'= hθ will be effective.

If ( i i i) does not hold apply the substitution 0 ' = ab - 3 bθ + a2 θ2; then 0 ' 3 +

cθ'2 +d = 0 where

Received February 10, 1954.Pacific J. Math. 5 (1955), 623-631

623

624 LEONARD TORNHEIM

Now 3 2 \\c since (b9 3) = 1. Also 3 4 \d. If 36|</, then the quantity 0 " = 0 7 9 s

satisfies the conditions of the lemma, where s is the largest integer for which

(s ,3) = 1, s \c, and s3\d. If 3 6 ^ use θ"= 0 7 3 s .

Essentially the following lemma is given by Sommer [2; p. 261],

LEMMA 2. 7ê integers of Q( 0), where 0 is described in Lemma 1, have a

basis over the integers given by

__β + <9 B2 Λ-aB + ( β + α ) 0 + θ2

ωx = 1, ω 2 = — , ω 3 =0 D 2 D

with B9 D9 Dι rational integers satisfying

(3) 3 β - f α =

( 4 ) 3B2 +2aB^0(D2Dι),

( 5 ) B3+aB2 +b^0(D3Df),

(6) -Δ = ό(4α 3 + 276) EO (D6D2),

and D9 D t are maximal subject to these conditions.

Proof. We shall first prove that D = 1. Let p be a prime dividing B and D.

By ( 3 ) , p also divides α. But then by ( 5 ) , p3\b, contradicting the choice of

0. Hence ( B , D) = 1.

From ( 3 ) and ( 4 ) , we have aB = 0 ( D ) . Therefore D \a. But by ( 3 ) , D = 3

or 1.

If D = 3, then 3fi> because from (5) we would get D \B. But then (6) con-

tradicts (i i i) of Lemma 1. Hence D = 1.

Therefore the problem is equivalent to determining the largest Di for which

there is a solution B satisfying (4), (5), (6), when D = 1. It is sufficient to

find solutions of these congruences with Dγ replaced by prime powers pΓ and

then Dγ will be their product. A value of B can be found from solutions modulo

pΓ by using the Chinese remainder theorem.

Thus we wish to determine the maximal value e of r for which there exists

a solution B of the simultaneous congruences

(7) B(3B

CUBIC FIELD 625

( 8 ) B3 + aB2 + b = 0{p2r),

( 9 ) - Δ = 6 ( 4 α 3 + 2 7 6 ) Ξ 0 ( p 2 r ) .

The power p e exists because of ( 9 ) ; in fact if p u | | Δ , then e <_s, where

5 = U / 2 ] .

Case I. ( p , 3 b) ~ 1. Then e - s. For, let B be a solution of

L = 3 β + 2α = 0 ( p s ) ;

hence ( 7 ) is satisfied. By ( 9 )

Now

L 3 - 3 α L 2

Ξ 0 ( p 2 s ) .

T h i s on e x p a n s i o n g i v e s

α ~ 4 α 3 = 0 ( p 2 s ) ,

w h i c h with t h e a b o v e formula s h o w s t h a t ( 8 ) i s s a t i s f i e d . T h u s ( 7 ) , ( 8 ) , ( 9 )

h o l d wi th r - s. H e n c e e >_ s . B u t s i n c e e < s we h a v e e = s .

Case II. p I 3 ό.

H i . ( p , 2 α ) = l . Then e = s . F o r , b y ( 9 ) , pu\\b. S i m p l y t a k e B=0(ps)

t o s e e t h a t ( 7 ) , ( 8 ) , ( 9 ) h o l d w i t h r = s.

Π i i . p \b9 p\a. Then e = 0 if p \\b and e = 1 = s - 1 if p2 \\b. N o t i c e t h a t

p \b by ( i i ) of L e m m a 1. F i r s t , if p \b9 t a k i n g B = = 0 ( p ) p r e s e n t s a s o l u t i o n

of t h e c o n g r u e n c e s wi th r = 1; t h u s e >_ 1. On the o t h e r h a n d , if e >. 1, t h e n

p\B by ( 8 ) ; s o t h a t p2 \b a g a i n by ( 8 ) . F i n a l l y , if e > 1 t h e n p 3 | ό by ( 8 )

s i n c e p | B by t h e p r e c e d i n g s e n t e n c e . T h i s i s a c o n t r a d i c t i o n to ( i i ) of L e m m a

1; h e n c e e < 1. It i s e a s y to s e e t h a t if pφ- 3, t h e n s = 1 w h e n p\\b a n d s — 2

when p 2 11 b. If p = 3, then s - 2 u n l e s s p 11 bf p 2 \ a and t h e n s = 3.

I l i i i . p = 3, p I α, p ^ 6 . N o t i c e t h a t t h e n s = 1 by ( 9 ) a n d ( i i i ) of L e m m a 1.

I Ι i i i ( l ) . 3 2 | α . Then e = 0 unless b = ± 1 ( 3 2 ) in which case e = 1. Now


e < s = 1. Furthermore, the fact that e = 1 if and only if 6 = ± 1 ( 3 ) i s a con-

sequence of (8) since only then does Z > 3 + 6 = 0 ( 3 2 ) have a solution for (3 ,6) = 1,

the solution being given b y β = = - 6 ( 3 ) ; ( 7 ) and ( 9 ) always hold with r = 1.

Iliii ( 2 ) . 3 11 α. Then e = 0, unless b + a = ± 1 ( 3 ), in which case e = 1.

That e < 1 is a consequence of ( 9 ) and ( i i i ) of Lemma 1. If r - 1, then ( 7 ) and

( 9 ) always hold and ( 8 ) has a solution if and only if£> + a s ± l ( 3 ). For,

if B sat is f ies ( 8 ) then 3 | # ; hence £ 2 = 1 ( 3 ) , aB2 + b = a + b ( 3 2 ) . But

B3 = ± 1 ( 3 2 ) so that α + 6 ^ - β 3 = + 1 ( 3 2 ) . Conversely, if a + b = + 1 ( 3 2 ) ,

take J δ Ξ - ( α + 6 ) Ξ ± l Ξ - 6 ( 3 ) ; then B3 + α δ 2 ύ E ί 1 + α + έ E O ( 3 2 ) .

Iliv. p = 2, 2 I 6, 2 | o . Define t and c by 2t \\ b, b = 2*c.

Hiv ( 1 ) . ί odd. From ( 7 ) , 2\B. In the expression on the left in ( 8 ) , there

is only one term, either aB2 or b, containing 2 to the lowest power. Hence

e < [ ί / 2 ] . But B = 0 ( 2 Γ ) with r = [ ί / 2 ] does provide a solution of the three

congruences. Hence e - [ ί / 2 ] Notice that e = 5 — 1 since u — t Λ-\ if ί = l

but u = t + 2 if ί > 1.

Iliv ( 2 ) . t = 2. Let 4 ^ | | ( 4 α 3 -h 27 6) , then u; > 1. Set 4 α 3 + 276 = 4 ^ . By

( 9 ) , e < + 1. Now e >_ w simply by replacing s by w in the solution of Case

I. It remains to determine when e = w + 1. Then from ( 7 ) , 2 | β and from ( 8 ) ,

2 2 | β . Also from ( 7 ) , 3β + 2a EΞ 0 ( 2 " ) ; that is , SB =~2a + 2wS. Now the

product of 27 with the congruence ( 8 ) gives

4 α 3 - 3 . 2 2 M ;

aS2 + 2 3 u ; S 3 + 276 s 0 ( 2 2 u ) + 2 ) .

Hence

2 ^ S 3 + / / ~ 3 α S 2 = 0 ( 2 2 ) .

If S = 0 ( 2 ) f then ff = 0 ( 4 ) f an impossibility. Hence S i s odd, S2 EE 1 ( 4 ) ,

S 3 = 5 ( 4 ) , and

2WS + f f + α = 0 ( 2 2 ) .

But since w > 1, we have 2^S = 2W ( 2 2 ) . Hence

CUBIC FIELD 627

(10) 2M; + / / + α = 0 ( 4 ) .

If w = I, then H = α 3 + 2 7 c = 0 ( 2 ) , a contradiction to (10) . Hence w > 1.

Conversely, if (10) is true, then all the congruences in this paragraph are

satisfied by taking S odd; that is, by taking for B a solution of

3B +2a^2w (2w+ι).

H e n c e e = w + 1 if and only if ( 1 0 ) i s s a t i s f i e d ; t h a t i s , H + a = 0 ( 4 ) . N o t i c e

from the definit ion of w that u = 2 + 2w; h e n c e s - w + 1.

Il iv ( 3 ) . t = 2v(v > 1 ) . From ( 9 ) , u = 2v 4-2; h e n c e e < s = v + 1. Now

β Ξ 0 ( 2 ^ ) y i e l d s a so lut ion of the c o n g r u e n c e s with r — v; h e n c e e >_ v We

determine when e = v + 1. T h e n from ( 7 ) , B i s even. Again from ( 7 ) e i ther

2 | | β or 2V I B. In the first c a s e v <_ 1 by ( 8 ) and t h i s i s a contrad ic t ion to

v > 1; h e n c e β ^ 2 V K. Now ( 7 ) h o l d s while ( 8 ) impl ies

23vK3 + a22vK2 +22vc^0 ( 2 2 t ; + 2 ) ,

which gives, since v > 1,

o χ 2 + C Ξ 0 ( 4 ) .

Thus K is odd and

a + c =0 ( 4 ) .

Conversely^ if this last congruence is satisfied and B is taken as a solution

of B Ξ= 2V ( 2 V + 1 ) , then β is a solution of ( 7 ) , ( 8 ) , and ( 9 ) .

These deductions are summarized in the following theorem.

THEOREM 1. Let θ satisfy the conditions of Lemma 1. A minimal basis

ofQ(θ)is

ω 1 = = l , ω2 = 0, ω 3 = {J3 2 + α β + ( β + a) θ + θ2 \/D,

where D is a product of prime powers ρe determined by the prime powers p

for which ( p 2 ) s | | Δ as described below and B is a common solution of the con-

gruences given below:

(1) If ( p , 3 6 ) = 1 , t h e n e = s a n d SB + 2 a 0 ( p e ) .

( 2 ) If p I a , p 11 b% t h e n e = 0 . , 4 Z s o e = s - 1 ί/ p / 3 cmc? e = s ~ 2 i / p = 3 .


( 3 ) If p\a9 p2\\b$ then e = 1 and B = 0 ( p e ) . Also e = s - 1 unless p = 3

am/ p I α α^ί/ ίλe z e = s - 2.

( 4 ) / / > I 36, ( p , 2 α ) = 1, then e = s and B = 0 ( p e ) .

( 5 ) 7/p = 3, 3 | α , 3^6, ίAerc e < 1 = s; and e = s ι/ cmc? orcZy i / έ + α Ξ + l ( 9 )

ami ίAera B = - 6 ( 3 ) .

( 6 ) / / p = 2, ( 2 , α ) = l , 2 * | | 6 αzzJ

( a ) if t is odd9 then e = s - 1 and B s O ( 2 e ) ;

( b ) if t = 2 ίΛerc e = s - 1 urc/ess H + a = 0 ( 4 ) , wΛere W = -

α/irf then e = s . ,4Zso 3 5 + 2α = 2 s " ι (2s).

( c ) if t > 2 and even, then e = 5 - 1 unless a + c=0 ( 4 ) , where c =6/2*,

and then e = 2. 4Zso B ^ 2 s " 1 ( 2 s ) .

The discriminant of (?(#) is Δ/D 2. It divides the discriminant Δ ( α ) of

every integer α of Q(Θ) and hence their g.c.d. G. The largest inessential

discriminant divisor F is the square root of the quotient G/(Δ/Z)2).

THEOREM 2. TΆe largest inessential discriminant divisor F is 1 except

it is 2 in Case 6b of Theorem 1 when

(11) //-3a + 2 e-U0(23)

αmZ in Case 6c when

(12) a + c + 2 e - 1 ^ 0 ( 2 3 ) .

Proof. The discriminant Δ(α) of an integer Cί = c\ ω\ + c 2 ω 2 + cz ω 3 can

be found from the formula

| α ι 7 | 2 Δ ( 0 ) ,

where the elements of the determinant | α t y | = |α, y (α) | are defined by

α1""ι = α a + ai2 0 + o i 3 02 ( j = 1, 2, 3 ) ,

Since the discriminant of α is unaltered by addition of a rational number, we

have

Δ(α) = Δ ( c 2 ω 2

where

CUBIC FIELD 629

+ C3{B + α ) / D ] # + (,

In computing β2 use the fact that θ3 = - aθ2 - b and θ4 = a2θ - bθ + ab. Also

since the first row of | α t y ( / 3 ) | is 1,0,0, any rational terms can be ignored.

Hence,

c3(B3+aB2+b) c,c2(3B2 + 2aB) c2cAW+a)/ Ί r ) \ | | 3 2 3 2 3 o

(13) | α , μ _ + _ + _ + c ».

Thu s

| α l 7 ( ω 3 ) | =D3

and

M , / x, ( 3 S + 2aS) ( 3 β + o )15 ) I αj.- ( ω 2 + ω 3 ) I - I o ί ; ( ω 3 ) | = + + 1.

D2 D

Now, since GD / Δ is the quotient of the g.c.d G of | α j y | 2 Δ by Δ/D , it

equals the g.c.d of \aη\ 2D2. Hence the inessential discriminant divisor F is

the g.c.d of I aij I D.

To find F we determine for each prime p the highest power p* which remains

in all the denominators of the | α j / ( c θ | expressed in their lowest terms. Then

F is the quotient of D divided by the product of these prime powers and thus F

is the product of all pe"K

In all c a s e s of Theorem 1 except in 5 when a + b = ± 1 ( 3 ), in 6b when

H + a = 0(22), and in 6c when α + c = 0 ( 2 2 ) , B may be chosen to satisfy

either

or

In these cases ( 1 5 ) implies, s ince its first term is then integral, that e = /

when p\a. But if p | a then p | b and s ince we need consider only e > 0 we have

Case 3 of Theorem 1. Then ( 1 4 ) with B E O ( p 3 e ) shows t h a t / = 1= e.


N e x t , in C a s e 5 w h e n b+a=±l ( 3 2 ) , 3 { β . If 3 2 | α , t h e n ( 1 5 ) i m p l i e s

t h a t / = 1 = e . B u t if 3 11 a t h e n a - 3 α i a n d aχ + 6 ^ 0 ( 3 ) by ( i i i ) of L e m m a 1.

Were / = 0 , t h e n B + 2a ι = 0 ( 3 ) b y ( 1 5 ) , w h i c h i m p l i e s B = ax ( 3 ) . B u t t h e n

B 3 + a B 2 + b = a 3 + b έ θ ( 3 ) ,

a contradiction to (8). Hence again f = e.

In both Cases 6b and 6c, 2 | β by (7). Now

2(3B + α )ω 3 ) I + |α jy(-ω 2 + ω 3 ) | - 2 | α ι ; ( ω 3 ) | =

D

Since 2 11 2 (3β + o), we have / > e - 1.

We now consider in particular Case 6b when ff + o = 0 (4) . Then 3β =

- 2a + 2e ιQ, where 0 is odd. Thus

2 7 ( β 3 +aB2 + 6 ) = 4 α 3 + 2 7 6 - 3 ρ 2 α 2 2 e - 2

+ ( ? 3 2 3 e - 3 .

Hence if f ~ e — 1, then

# - 3 α + 2 e - 1 = 0 ( 2 3 )

by (14), and if this is satisfied then / = e - 1. For, the first term in (13) has

numerator divisible by 2 2 e + 1 , and 2 e | | ( 3 β 2 + 2aB) and 2° | | ( 3 β + o ) so that

2 e + 1 | [ c 2 c 3

2 ( 3 β 2 + 2 α β ) + O C 2

2

C 3 ( 3 β + α ) ] .

Hence in lowest terms \a{j \ has a denominator divisible by no power of p greater

than e — 1.

We finally discuss Case 6c when α + c = 0 (4) . Then β = 2 6 " 1 + C2 e, where

we may assume that 2 e + 2 | C, and b = 2 2 ^ e " ι ^c. Hence

If / = e - 1, then by (14) this expression must be = 0 ( 2 2 e + 1 ) , so that

If this is satisfied then / = e - 1 because the first term of (13) has numerator

divisible by 2 2 e + 1 , and 2 e 11 ( 3β 2 + 2αβ) and 2° 11 (3β + a) so that

CUBIC FIELD 631

REFERENCES

1. A.A.Albert, A determination of the integers of all cubic fields, Ann. of Math.,31 (1930), 550-566.

2. J. Sommer, Vorlesungen ilber Zahlentheorie, Berlin, 1907.

3. E. v. Zylinski, Zur Theorie der aus serwe sentliche D is krminantenteiler alge-braischer Korper, Math. Ann. 73 (1913), 273-274.

UNIVERSITY OF MICHIGAN

ON EIGENVALUES OF SUMS OF NORMAL MATRICES

H E L M U T W I E L A N D T

1. Problem, notations, results. A well-known theorem due essential ly to

Bendixson [ l , Theorem I I ] s ta tes that if X and Y are hermitian nxn matrices

with eigenvalues

ζγ <_ ζ2 <_ < ζn and η ι <C 77 2 < <_ η^ ,

then every eigenvalue λ of X + iY is contained in the rectangle

What is the exact range of λ, for given ζv and ηv? We shall solve the following

slightly more general problem, referring to normal instead of hermitian matrices.

Let Cί i, , Cί j, βι9 9 β be given complex numbers. Describe geometrically

the set A of all numbers λ which may occur as eigenvalues of A + B, where A

and B run over all normal nxn matrices with eigenvalues OC i, , CXn and

βί9 *, βn respectively.

To state the results concisely let us denote by the terms circular region and

hyperbolic region every set of complex numbers ζ + iη which may be described,

using some real constants α, bf c9 d9 by

(1) aξ+bη + c{ξ2± η2)+d > 0 .

where + refers to the circular, - to the hyperbolic case. We denote by \ Qiv\

and \βv\ύie sets whose elements are Cίi, ,(Xπ and β ^ , / ^ respectively.

For every two sets Γ, Δ of complex numbers we denote by Γ 4- Δ the set whose

elements are all γ 4- δ, where y G Γ , 8 € Δ. Our main result is

THEOREM 1. // Oli, •••, 0Cπ, βι9 ,βn are arbitrary complex numbers,

then the set A defined above can be represented as an intersection:

Received April 16, 1954. This paper was prepared under a National Bureau of Stand-ards contract with the American University.

Pacific J. Math. 5 (1955), 633-638

633

634 HELMUT WIELANDT

where Γ runs over all circular regions which contain \ βv\

In the special case considered by Bendixson the result may be simplified

as follows.

THEOREM 2. If &ι, , (Xn are real and /31, , βn purely imaginary, then

Λ = Π ΔΔ

where Δ runs over all hyperbolic regions which contain \<XV\ + ί βp\.

2. Proofs. We recall the following theorem [ 3 , Theorem 2 ] : Let λ 1 ? , λΛ

be complex numbers, y and z complex n xl matrices, y*y = 1. Denote by

M{y,z) the point in real 3-space with rectangular coordinates [\iy*z, & y*z9

z*z ] and by P{ζ), for every complex number ζ, the point [ R £ , &ζ, \ζ\2]

If, and only if, the convex closure of the n points P(λv) contains M(y9z)9 then

there exists a normal nxn matrix L with eigenvalues λ-u 9 λn such that

Using the notations introduced in § 1 we prove for arbitrary CCV, βv:

LEMMA 1. Let ζ be a complex number. Then ( E Λ if, and only if, the con-

vex closure C of P ( d i ) , , P ( CLn) has a point in common with the convex

closure C ' of P ( ζ - βχ ), ,P{ζ- β n ) .

Proof, ( a ) Let ( G Λ , Then there are normal matrices A, B with spectra

&1> * * •> an a n d βt9 * '> βn

s u c n that

{A +B-ζI)y=0

for some normalized vector y. Putting

we conclude from the necessity part of the theorem quoted above that M{y,z)

is a common point of C and C".

(b) On the other hand, let there be some point

Then iΛ2+^2 < ,» hence we have N = M(y,z) for some vectors y, z such

ON EIGENVALUES OF SUMS OF NORMAL MATRICES 6 3 5

that y*y = 1. By the sufficiency part of the theorem quoted above there are

normal matrices A9 B with eigenvalues (Xv, βv^ ζ- βv such that Ay = z = By.

Define B - ζl - B. Then B has the eigenvalues βv and satisfies Ay = ( ζl — B) y,

hence ζ is an eigenvalue of A + B (with γ as a corresponding eigenvector).

We transform the "three-dimensional" Lemma 1 into a "two-dimensional"

form.

LEMMA 2. Let ζ be a complex number. Then ζ<£A if, and only if, there

exists a circle or a straight line separating CX 1 ? , dn from ζ— βι, , ζ— β .

Proof. From Lemma 1 we know that ζ£ A if, and only if, there exists a

plane separating C from C', that is, separating

P ( α i ) , . . , P ( α Λ )

from

Piζ-βJ,..., P(ζ-βn).

T h i s m e a n s t h a t ζ£A if, a n d o n l y if, t h e r e a r e r e a l c o n s t a n t s a9b9c9d s u c h

t h a t for v = 1, , n

a R α v + b&av + c\av\2 + d > 0 ,

( 2 )

& β v ) + c \ ζ - β v \ 2 +d < o .

This proves Lemma 2. We turn to the proof of Theorem 1.

(a) Let ζ G Λ, and let Γ be any circular region containing /3 1 ? 9 βn

We have to show that £ E { α v ϊ + Γ. Now ζ— Γ is a circular region containing

ζ — βχ9 , ζ — βn By Lemma 2 there is at least one Cίp such that

(b) Let ζ£ A, Then Theorem 1 claims that there is a circular region Γ

containing \βv\ such that ζfl:\CLv} +T. Indeed, interchanging A and β in

Lemma 2 we see that there is a circular region Γ containing β 9 , β , but

none of the ζ— CCV

Theorem 1 being proved, we turn to Bendixson's case where QLV = ξv is

real, and βv-ir]v with ηy real. We have to show, for any complex number ζ,

636 HELMUT WIELANDT

that ζ jέ Λ if, and only if, there is a hyperbolic region which contains all points

ζn + iΉvi but does not contain ζ. Since this statement obviously is not affected

by a translation applied simultaneously to the ζ + iη and to ζ, we may assume

without loss of generality that ζ- 0.

From (2) we know that 0 ^ Λ if, and only if, there are real numbers α, b, c9 d

such that

aξv + cξl+d > 0

(3) ( v = l , . . . , n ) .

-bηv+cηl + d > 0

This condition is equivalent to the existence of real numbers, α, b9 c such that

(4) aξμ+bηv + c(ξ^-r^)>0 (μ,v=l,...,n).

This inequality is equivalent to the existence of a hyperbolic region containing

all points ζ + i ηv, but not 0.

3. Remarks, (a) It is seen from Theorem 1 that in the determination of

Λ only the distinct (Xv and the distinct βv matter. Multiplicities are of no im-

portance.

(b) If ί α ^ i C ίcίvίand ί/3 piC { βv\, then ΛC Λ.

Proof. For every λ G Λ there are normal r x r matrices A and B with eigen-

values dp and βp such that λ is an eigenvalue of A + B. Define

φ,), B.[\)\ <*„/ \ A,/

Then λ is an eigenvalue of A + B. The eigenvalues of A and B coincide with

CCi, , 0Cn and βγ, , βn except for the multiplicities; hence λ £ Λ by ( a ) .

( c ) Λ is a closed bounded set, since the se t s ίCί^S + Γ are closed and

there is a bounded circular region Γ containing \βv\.

( d ) In Bendixson's case every connected component of Λ is simply con-

nected.

Proof. By Theorem 2 every point of the complement Λ* of Λ is the end

ON EIGENVALUES OF SUMS OF NORMAL MATRICES 637

point of some half line which is entirely contained in A*. Moreover by (c) , A*

contains the exterior of some circle. Hence A* is connected.

(e) If n — 2 in Bendixson's case then Theorem 2 implies that A is the

intersection of Bendixson's rectangle with the rectangular hyperbola passing

through the vertices of that rectangle. This result has been previously ob-

tained by W. V. Parker [2, Theorem 1 ].

(f) The foregoing remarks lead to a simple procedure for constructing

A in Bendixson's case. Let (with a slight change of notation) ξγ < ζ2 <•••

< ζ be the distinct eigenvalues of the hermitian n x n matrix X, and let

?7t < η2 < < η^ be the distinct eigenvalues of the hermitian nxn matrix

y. We define p.μ ( μ - 1,

which passes through.

, m - 1) to be that part of the rectangular hyperbola

which lies in the rectangle with these vertices. Similarly we define H^ί K =

1, •••,&--1) interchanging the role of the <f's and the 77's. Then (b) and (e)

show that ΞμC^ A and II/< C A. It is easily seen that the union of all Ξ μ ' s and

H/<'s consists of one or more closed Jordan curves each of which is contained

in the closed exterior of every other curve, and that no point exterior to all

curves belongs to A. On the other hand, the interior of every curve is con-

tained in A, by (d) . Hence λ is the largest bounded region whose boundary

is the union of3i , , *Bm-1 , H l 5 , H^. t .

As an example we construct the range A of the eigenvalues of X 4- iY where

X and y are hermitian matrices whose eigenvalues are 0, 1,4,8 and 0,2,3

(with arbitrary positive multiplicities).

638 HELMUT WIELANDT

REFERENCES

1. I. Bendixson, Sur les racines d'une equation fondamentale, Acta Math. 25 (1902),359-365.

2. W.V.Parker, Characteristic roots of matrices, Amer. Math. Monthly 60 (1953),247-250.

3. H. Wielandt, Die E ins chlies sung der Eigenwerte normaler Matrίzen, Math. Ann.121 (1949), 234-241.

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Pacific Journal of MathematicsVol. 5, No. 4 December, 1955

Richard Horace Battin, Note on the “Evaluation of an integral occurring inservomechanism theory” . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 481

Frank Herbert Brownell, III, An extension of Weyl’s asymptotic law foreigenvalues . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 483

Wilbur Eugene Deskins, On the homomorphisms of an algebra ontoFrobenius algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 501

James Michael Gardner Fell, The measure ring for a cube of arbitrarydimension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 513

Harley M. Flanders, The norm function of an algebraic field extension.II . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 519

Dieter Gaier, On the change of index for summable series . . . . . . . . . . . . . . . . . 529Marshall Hall and Lowell J. Paige, Complete mappings of finite groups . . . . . 541Moses Richardson, Relativization and extension of solutions of irreflexive

relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 551Peter Scherk, An inequality for sets of integers . . . . . . . . . . . . . . . . . . . . . . . . . . . 585W. R. Scott, On infinite groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 589A. Seidenberg, On homogeneous linear differential equations with arbitrary

constant coefficients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 599Victor Lenard Shapiro, Cantor-type uniqueness of multiple trigonometric

integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 607Leonard Tornheim, Minimal basis and inessential discriminant divisors for

a cubic field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 623Helmut Wielandt, On eigenvalues of sums of normal matrices . . . . . . . . . . . . . 633

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athematics

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