Field Extensions

Field extensionsFrom Wikipedia, the free encyclopedia

Chapter 1

Abelian extension

In abstract algebra, an abelian extension is a Galois extension whose Galois group is abelian. When the Galois groupis also cyclic, the extension is also called a cyclic extension. Going in the other direction, a Galois extension is calledsolvable if its Galois group is solvable, i.e., if the group can be decomposed into a series of normal extensions of anabelian group.Every finite extension of a finite field is a cyclic extension.Class field theory provides detailed information about the abelian extensions of number fields, function fields ofalgebraic curves over finite fields, and local fields.There are two slightly different definitions of the term cyclotomic extension. It can mean either an extension formedby adjoining roots of unity to a field, or a subextension of such an extension. The cyclotomic fields are examples. Acyclotomic extension, under either definition, is always abelian.If a field K contains a primitive n-th root of unity and the n-th root of an element of K is adjoined, the resultingso-called Kummer extension is an abelian extension (if K has characteristic p we should say that p doesn't divide n,since otherwise this can fail even to be a separable extension). In general, however, the Galois groups of n-th rootsof elements operate both on the n-th roots and on the roots of unity, giving a non-abelian Galois group as semi-directproduct. The Kummer theory gives a complete description of the abelian extension case, and the Kronecker–Webertheorem tells us that if K is the field of rational numbers, an extension is abelian if and only if it is a subfield of a fieldobtained by adjoining a root of unity.There is an important analogy with the fundamental group in topology, which classifies all covering spaces of a space:abelian covers are classified by its abelianisation which relates directly to the first homology group.

1.1 References• Kuz'min, L.V. (2001), “cyclotomic extension”, inHazewinkel, Michiel, Encyclopedia ofMathematics, Springer,ISBN 978-1-55608-010-4

2

Chapter 2

Algebraic closure

For other uses, see Closure (disambiguation).

In mathematics, particularly abstract algebra, an algebraic closure of a field K is an algebraic extension of K that isalgebraically closed. It is one of many closures in mathematics.Using Zorn’s lemma, it can be shown that every field has an algebraic closure,[1][2][3] and that the algebraic closureof a field K is unique up to an isomorphism that fixes every member of K. Because of this essential uniqueness, weoften speak of the algebraic closure of K, rather than an algebraic closure of K.The algebraic closure of a field K can be thought of as the largest algebraic extension of K. To see this, note that if Lis any algebraic extension of K, then the algebraic closure of L is also an algebraic closure of K, and so L is containedwithin the algebraic closure of K. The algebraic closure of K is also the smallest algebraically closed field containingK, because ifM is any algebraically closed field containing K, then the elements ofM that are algebraic over K forman algebraic closure of K.The algebraic closure of a field K has the same cardinality as K if K is infinite, and is countably infinite if K is finite.[3]

2.1 Examples• The fundamental theorem of algebra states that the algebraic closure of the field of real numbers is the field ofcomplex numbers.

• The algebraic closure of the field of rational numbers is the field of algebraic numbers.

• There are many countable algebraically closed fields within the complex numbers, and strictly containing thefield of algebraic numbers; these are the algebraic closures of transcendental extensions of the rational numbers,e.g. the algebraic closure of Q(π).

• For a finite field of prime power order q, the algebraic closure is a countably infinite field that contains a copyof the field of order qn for each positive integer n (and is in fact the union of these copies).[4]

2.2 Existence of an algebraic closure and splitting fields

Let S = {fλ|λ ∈ Λ} be the set of all monic irreducible polynomials in K[x]. For each fλ ∈ S , introduce newvariables uλ,1, . . . , uλ,d where d = degree(fλ) . Let R be the polynomial ring over K generated by uλ,i for all λ ∈ Λand all i ≤ degree(fλ) . Write

fλ −d∏

i=1

(x− uλ,i) =d−1∑j=0

rλ,j · xj ∈ R[x]

3

4 CHAPTER 2. ALGEBRAIC CLOSURE

with rλ,j ∈ R . Let I be the ideal in R generated by the rλ,j . Since I is strictly smaller than R, Zorn’s lemmaimplies that there exists a maximal idealM in R that contains I. Now the field R/M is an algebraic closure of K: everypolynomial fλ splits as the product of the x− (uλ,i +M) .The same proof also shows that for any subset S of K[x], there exists a splitting field of S over K.

2.3 Separable closure

An algebraic closure Kalg of K contains a unique separable extension Ksep of K containing all (algebraic) separableextensions of K within Kalg. This subextension is called a separable closure of K. Since a separable extension ofa separable extension is again separable, there are no finite separable extensions of Ksep, of degree > 1. Saying thisanother way, K is contained in a separably-closed algebraic extension field. It is unique (up to isomorphism).[5]

The separable closure is the full algebraic closure if and only if K is a perfect field. For example, if K is a field ofcharacteristic p and if X is transcendental over K,K(X)( p

√X) ⊃ K(X) is a non-separable algebraic field extension.

In general, the absolute Galois group of K is the Galois group of Ksep over K.[6]

2.4 See also• Algebraically closed field

• Algebraic extension

• Puiseux expansion

2.5 References[1] McCarthy (1991) p.21

[2] M. F. Atiyah and I. G. Macdonald (1969). Introduction to commutative algebra. Addison-Wesley publishing Company. pp.11-12.

[3] Kaplansky (1972) pp.74-76

[4] Brawley, Joel V.; Schnibben, George E. (1989), “2.2 The Algebraic Closure of a Finite Field”, Infinite Algebraic Extensionsof Finite Fields, Contemporary Mathematics 95, American Mathematical Society, pp. 22–23, ISBN 978-0-8218-5428-0,Zbl 0674.12009.

[5] McCarthy (1991) p.22

[6] Fried, Michael D.; Jarden, Moshe (2008). Field arithmetic. Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge11 (3rd ed.). Springer-Verlag. p. 12. ISBN 978-3-540-77269-9. Zbl 1145.12001.

• Kaplansky, Irving (1972). Fields and rings. Chicago lectures in mathematics (Second ed.). University ofChicago Press. ISBN 0-226-42451-0. Zbl 1001.16500.

• McCarthy, Paul J. (1991). Algebraic extensions of fields (Corrected reprint of the 2nd ed.). New York: DoverPublications. Zbl 0768.12001.

Chapter 3

Algebraic extension

In abstract algebra, a field extension L/K is called algebraic if every element of L is algebraic over K, i.e. if everyelement of L is a root of some non-zero polynomial with coefficients in K. Field extensions that are not algebraic, i.e.which contain transcendental elements, are called transcendental.For example, the field extensionR/Q, that is the field of real numbers as an extension of the field of rational numbers,is transcendental, while the field extensionsC/R andQ(√2)/Q are algebraic, whereC is the field of complex numbers.All transcendental extensions are of infinite degree. This in turn implies that all finite extensions are algebraic.[1] Theconverse is not true however: there are infinite extensions which are algebraic. For instance, the field of all algebraicnumbers is an infinite algebraic extension of the rational numbers.If a is algebraic over K, then K[a], the set of all polynomials in a with coefficients in K, is not only a ring but a field:an algebraic extension of K which has finite degree over K. The converse is true as well, if K[a] is a field, then a isalgebraic over K. In the special case where K =Q is the field of rational numbers, Q[a] is an example of an algebraicnumber field.A field with no nontrivial algebraic extensions is called algebraically closed. An example is the field of complexnumbers. Every field has an algebraic extension which is algebraically closed (called its algebraic closure), but provingthis in general requires some form of the axiom of choice.An extension L/K is algebraic if and only if every sub K-algebra of L is a field.

3.1 Properties

The class of algebraic extensions forms a distinguished class of field extensions, that is, the following three propertieshold:[2]

1. If E is an algebraic extension of F and F is an algebraic extension of K then E is an algebraic extension of K.

2. If E and F are algebraic extensions of K in a common overfield C, then the compositum EF is an algebraicextension of K.

3. If E is an algebraic extension of F and E>K>F then E is an algebraic extension of K.

These finitary results can be generalized using transfinite induction:

1. The union of any chain of algebraic extensions over a base field is itself an algebraic extension over the samebase field.

This fact, together with Zorn’s lemma (applied to an appropriately chosen poset), establishes the existence of algebraicclosures.

5

6 CHAPTER 3. ALGEBRAIC EXTENSION

3.2 Generalizations

Main article: Substructure

Model theory generalizes the notion of algebraic extension to arbitrary theories: an embedding ofM into N is calledan algebraic extension if for every x in N there is a formula p with parameters inM, such that p(x) is true and the set

{y ∈ N

∣∣∣p(y)}is finite. It turns out that applying this definition to the theory of fields gives the usual definition of algebraic extension.The Galois group of N overM can again be defined as the group of automorphisms, and it turns out that most of thetheory of Galois groups can be developed for the general case.

3.3 See also• Integral element

• Lüroth’s theorem

• Galois extension

• Separable extension

• Normal extension

3.4 Notes[1] See also Hazewinkel et al. (2004), p. 3.

[2] Lang (2002) p.228

3.5 References• Hazewinkel, Michiel; Gubareni, Nadiya; Gubareni, Nadezhda Mikhaĭlovna; Kirichenko, Vladimir V. (2004),Algebras, rings and modules 1, Springer, ISBN 1-4020-2690-0

• Lang, Serge (1993), “V.1:Algebraic Extensions”, Algebra (Third ed.), Reading, Mass.: Addison-Wesley Pub.Co., pp. 223ff, ISBN 978-0-201-55540-0, Zbl 0848.13001

• McCarthy, Paul J. (1991) [corrected reprint of 2nd edition, 1976], Algebraic extensions of fields, New York:Dover Publications, ISBN 0-486-66651-4, Zbl 0768.12001

• Roman, Steven (1995), Field Theory, GTM 158, Springer-Verlag, ISBN 9780387944081

• Rotman, Joseph J. (2002), Advanced Modern Algebra, Prentice Hall, ISBN 9780130878687

Chapter 4

Characteristic (algebra)

In mathematics, the characteristic of a ring R, often denoted char(R), is defined to be the smallest number of timesone must use the ring’s multiplicative identity element (1) in a sum to get the additive identity element (0); the ring issaid to have characteristic zero if this sum never reaches the additive identity.That is, char(R) is the smallest positive number n such that

1 + · · ·+ 1︸ ︷︷ ︸nsummands

= 0

if such a number n exists, and 0 otherwise.The characteristic may also be taken to be the exponent of the ring’s additive group, that is, the smallest positive nsuch that

a+ · · ·+ a︸ ︷︷ ︸nsummands

= 0

for every element a of the ring (again, if n exists; otherwise zero). Some authors do not include the multiplicativeidentity element in their requirements for a ring (see ring), and this definition is suitable for that convention; otherwisethe two definitions are equivalent due to the distributive law in rings.

4.1 Other equivalent characterizations

• The characteristic is the natural number n such that nZ is the kernel of a ring homomorphism from Z to R;

• The characteristic is the natural number n such that R contains a subring isomorphic to the factor ring Z/nZ,which would be the image of that homomorphism.

• When the non-negative integers {0, 1, 2, 3, . . . } are partially ordered by divisibility, then 1 is the smallestand 0 is the largest. Then the characteristic of a ring is the smallest value of n for which n · 1 = 0. If nothing“smaller” (in this ordering) than 0 will suffice, then the characteristic is 0. This is the right partial orderingbecause of such facts as that char A × B is the least common multiple of char A and char B, and that no ringhomomorphism ƒ : A→ B exists unless char B divides char A.

• The characteristic of a ring R is n ∈ {0, 1, 2, 3, . . . } precisely if the statement ka = 0 for all a ∈ R implies nis a divisor of k.

The requirements of ring homomorphisms are such that there can be only one homomorphism from the ring ofintegers to any ring; in the language of category theory, Z is an initial object of the category of rings. Again thisfollows the convention that a ring has a multiplicative identity element (which is preserved by ring homomorphisms).

7

8 CHAPTER 4. CHARACTERISTIC (ALGEBRA)

4.2 Case of rings

If R and S are rings and there exists a ring homomorphism R→ S, then the characteristic of S divides the characteristicof R. This can sometimes be used to exclude the possibility of certain ring homomorphisms. The only ring withcharacteristic 1 is the trivial ring which has only a single element 0 = 1. If a non-trivial ring R does not have any zerodivisors, then its characteristic is either 0 or prime. In particular, this applies to all fields, to all integral domains, andto all division rings. Any ring of characteristic 0 is infinite.The ring Z/nZ of integers modulo n has characteristic n. If R is a subring of S, then R and S have the same char-acteristic. For instance, if q(X) is a prime polynomial with coefficients in the field Z/pZ where p is prime, then thefactor ring (Z/pZ)[X]/(q(X)) is a field of characteristic p. Since the complex numbers contain the rationals, theircharacteristic is 0.A Z/nZ-algebra is equivalently a ring whose characteristic divides n.If a commutative ring R has prime characteristic p, then we have (x + y)p = xp + yp for all elements x and y in R –the "freshman’s dream" holds for power p.The map

f(x) = xp

then defines a ring homomorphism

R→ R.

It is called the Frobenius homomorphism. If R is an integral domain it is injective.

4.3 Case of fields

As mentioned above, the characteristic of any field is either 0 or a prime number. A field of non-zero characteristicis called a field of finite characteristic or a field of positive characteristic.For any fieldF, there is aminimal subfield, namely the prime field, the smallest subfield containing 1F. It is isomorphiceither to the rational number field Q, or a finite field of prime order, Fp; the structure of the prime field and thecharacteristic each determine the other. Fields of characteristic zero have the most familiar properties; for practicalpurposes they resemble subfields of the complex numbers (unless they have very large cardinality, that is; in fact, anyfield of characteristic zero and cardinality at most continuum is isomorphic to a subfield of complex numbers).[1] Thep-adic fields or any finite extension of them are characteristic zero fields, much applied in number theory, that areconstructed from rings of characteristic pk, as k→∞.For any ordered field, as the field of rational numbers Q or the field of real numbers R, the characteristic is 0. Thus,number fields and the field of complex numbers C are of characteristic zero. Actually, every field of characteristiczero is the quotient field of a ring Q[X]/P where X is a set of variables and P a set of polynomials in Q[X]. Thefinite field GF(pn) has characteristic p. There exist infinite fields of prime characteristic. For example, the field ofall rational functions over Z/pZ, the algebraic closure of Z/pZ or the field of formal Laurent series Z/pZ((T)). Thecharacteristic exponent is defined similarly, except that it is equal to 1 if the characteristic is zero; otherwise it has thesame value as the characteristic.[2]

The size of any finite ring of prime characteristic p is a power of p. Since in that case it must contain Z/pZ it mustalso be a vector space over that field and from linear algebra we know that the sizes of finite vector spaces over finitefields are a power of the size of the field. This also shows that the size of any finite vector space is a prime power. (Itis a vector space over a finite field, which we have shown to be of size pn. So its size is (pn)m = pnm.)

4.4 References[1] Enderton, Herbert B. (2001),AMathematical Introduction to Logic (2nd ed.), Academic Press, p. 158, ISBN9780080496467.

Enderton states this result explicitly only for algebraically closed fields, but also describes a decomposition of any field asan algebraic extension of a transcendental extension of its prime field, from which the result follows immediately.

4.4. REFERENCES 9

[2] “Field Characteristic Exponent”. Wolfram Mathworld. Wolfram Research. Retrieved May 27, 2015.

• Neal H. McCoy (1964, 1973) The Theory of Rings, Chelsea Publishing, page 4.

Chapter 5

Degree of a field extension

In mathematics, more specifically field theory, the degree of a field extension is a rough measure of the “size” ofthe field extension. The concept plays an important role in many parts of mathematics, including algebra and numbertheory — indeed in any area where fields appear prominently.

5.1 Definition and notation

Suppose that E/F is a field extension. Then E may be considered as a vector space over F (the field of scalars). Thedimension of this vector space is called the degree of the field extension, and it is denoted by [E:F].The degree may be finite or infinite, the field being called a finite extension or infinite extension accordingly. Anextension E/F is also sometimes said to be simply finite if it is a finite extension; this should not be confused with thefields themselves being finite fields (fields with finitely many elements).The degree should not be confused with the transcendence degree of a field; for example, the field Q(X) of rationalfunctions has infinite degree over Q, but transcendence degree only equal to 1.

5.2 The multiplicativity formula for degrees

Given three fields arranged in a tower, say K a subfield of L which is in turn a subfield ofM, there is a simple relationbetween the degrees of the three extensions L/K, M/L and M/K:

[M : K] = [M : L] · [L : K].

In other words, the degree going from the “bottom” to the “top” field is just the product of the degrees going fromthe “bottom” to the “middle” and then from the “middle” to the “top”. It is quite analogous to Lagrange’s theorem ingroup theory, which relates the order of a group to the order and index of a subgroup — indeed Galois theory showsthat this analogy is more than just a coincidence.The formula holds for both finite and infinite degree extensions. In the infinite case, the product is interpreted in thesense of products of cardinal numbers. In particular, this means that if M/K is finite, then both M/L and L/K arefinite.IfM/K is finite, then the formula imposes strong restrictions on the kinds of fields that can occur betweenM andK, viasimple arithmetical considerations. For example, if the degree [M:K] is a prime number p, then for any intermediatefield L, one of two things can happen: either [M:L] = p and [L:K] = 1, in which case L is equal to K, or [M:L] = 1 and[L:K] = p, in which case L is equal toM. Therefore there are no intermediate fields (apart fromM and K themselves).

10

5.2. THE MULTIPLICATIVITY FORMULA FOR DEGREES 11

5.2.1 Proof of the multiplicativity formula in the finite case

Suppose that K, L andM form a tower of fields as in the degree formula above, and that both d = [L:K] and e = [M:L]are finite. This means that we may select a basis {u1, ..., ud} for L over K, and a basis {w1, ..., we} forM over L. Wewill show that the elements umwn, for m ranging through 1, 2, ..., d and n ranging through 1, 2, ..., e, form a basis forM/K; since there are precisely de of them, this proves that the dimension of M/K is de, which is the desired result.First we check that they spanM/K. If x is any element ofM, then since the wn form a basis forM over L, we can findelements an in L such that

x =e∑

n=1

anwn = a1w1 + · · ·+ aewe.

Then, since the um form a basis for L over K, we can find elements bm,n in K such that for each n,

an =d∑

m=1

bm,num = b1,nu1 + · · ·+ bd,nud.

Then using the distributive law and associativity of multiplication in M we have

x =e∑

n=1

(d∑

m=1

bm,num

)wn =

e∑n=1

d∑m=1

bm,n(umwn),

which shows that x is a linear combination of the umwn with coefficients from K; in other words they spanM over K.Secondly we must check that they are linearly independent over K. So assume that

0 =e∑

n=1

d∑m=1

bm,n(umwn)

for some coefficients bm,n in K. Using distributivity and associativity again, we can group the terms as

0 =e∑

n=1

(d∑

m=1

bm,num

)wn,

and we see that the terms in parentheses must be zero, because they are elements of L, and the wn are linearlyindependent over L. That is,

0 =d∑

m=1

bm,num

for each n. Then, since the bm,n coefficients are in K, and the um are linearly independent over K, we must have thatbm,n = 0 for all m and all n. This shows that the elements umwn are linearly independent over K. This concludes theproof.

5.2.2 Proof of the formula in the infinite case

In this case, we start with bases uα and wᵦ of L/K and M/L respectively, where α is taken from an indexing set A,and β from an indexing set B. Using an entirely similar argument as the one above, we find that the products uαwᵦform a basis for M/K. These are indexed by the cartesian product A × B, which by definition has cardinality equal tothe product of the cardinalities of A and B.

12 CHAPTER 5. DEGREE OF A FIELD EXTENSION

5.3 Examples• The complex numbers are a field extension over the real numbers with degree [C:R] = 2, and thus there are nonon-trivial fields between them.

• The field extension Q(√2, √3), obtained by adjoining √2 and √3 to the field Q of rational numbers, has degree4, that is, [Q(√2, √3):Q] = 4. The intermediate field Q(√2) has degree 2 over Q; we conclude from themultiplicativity formula that [Q(√2, √3):Q(√2)] = 4/2 = 2.

• The finite field (Galois field) GF(125) = GF(53) has degree 3 over its subfield GF(5). More generally, if p isa prime and n, m are positive integers with n dividing m, then [GF(pm):GF(pn)] = m/n.

• The field extensionC(T)/C, whereC(T) is the field of rational functions overC, has infinite degree (indeed it isa purely transcendental extension). This can be seen by observing that the elements 1, T, T2, etc., are linearlyindependent over C.

• The field extension C(T2) also has infinite degree over C. However, if we view C(T2) as a subfield of C(T),then in fact [C(T):C(T2)] = 2. More generally, if X and Y are algebraic curves over a field K, and F : X → Yis a surjective morphism between them of degree d, then the function fields K(X) and K(Y) are both of infinitedegree over K, but the degree [K(X):K(Y)] turns out to be equal to d.

5.4 Generalization

Given two division rings E and F with F contained in E and the multiplication and addition of F being the restrictionof the operations in E, we can consider E as a vector space over F in two ways: having the scalars act on the left,giving a dimension [E:F] , and having them act on the right, giving a dimension [E:F]ᵣ. The two dimensions need notagree. Both dimensions however satisfy a multiplication formula for towers of division rings; the proof above appliesto left-acting scalars without change.

5.5 References• page 215, Jacobson, N. (1985). Basic Algebra I. W. H. Freeman and Company. ISBN 0-7167-1480-9. Proofof the multiplicativity formula.

• page 465, Jacobson, N. (1989). Basic Algebra II. W. H. Freeman and Company. ISBN 0-7167-1933-9. Brieflydiscusses the infinite dimensional case.

Chapter 6

Dual basis in a field extension

In mathematics, the linear algebra concept of dual basis can be applied in the context of a finite extension L/K, byusing the field trace. This requires the property that the field trace TrL/K provides a non-degenerate quadratic formover K. This can be guaranteed if the extension is separable; it is automatically true if K is a perfect field, and hencein the cases where K is finite, or of characteristic zero.A dual basis is not a concrete basis like the polynomial basis or the normal basis; rather it provides a way of using asecond basis for computations.Consider two bases for elements in a finite field, GF(pm):

B1 = α0, α1, . . . , αm−1

and

B2 = γ0, γ1, . . . , γm−1

then B2 can be considered a dual basis of B1 provided

Tr(αi · γj) ={0, if i ̸= j1, otherwise

Here the trace of a value in GF(pm) can be calculated as follows:

Tr(β) =m−1∑i=0

βpi

Using a dual basis can provide a way to easily communicate between devices that use different bases, rather than havingto explicitly convert between bases using the change of bases formula. Furthermore, if a dual basis is implementedthen conversion from an element in the original basis to the dual basis can be accomplished with a multiplication bythe multiplicative identity (usually 1).

13

Chapter 7

Field extension

In abstract algebra, field extensions are the main object of study in field theory. The general idea is to start with abase field and construct in some manner a larger field that contains the base field and satisfies additional properties.For instance, the set Q(√2) = {a + b√2 | a, b ∈ Q} is the smallest extension of Q that includes every real solution tothe equation x2 = 2.

7.1 Definitions

Let L be a field. A subfield of L is a subset K of L that is closed under the field operations of L and under takinginverses in L. In other words, K is a field with respect to the field operations inherited from L. The larger field L isthen said to be an extension field of K. To simplify notation and terminology, one says that L / K (read as "L overK") is a field extension to signify that L is an extension field of K.If L is an extension of F which is in turn an extension ofK, then F is said to be an intermediate field (or intermediateextension or subextension) of the field extension L / K.Given a field extension L / K and a subset S of L, the smallest subfield of L which contains K and S is denoted byK(S)—i.e. K(S) is the field generated by adjoining the elements of S to K. If S consists of only one element s, K(s) isa shorthand for K({s}). A field extension of the form L = K(s) is called a simple extension and s is called a primitiveelement of the extension.Given a field extension L / K, the larger field L can be considered as a vector space over K. The elements of L arethe “vectors” and the elements of K are the “scalars”, with vector addition and scalar multiplication obtained fromthe corresponding field operations. The dimension of this vector space is called the degree of the extension and isdenoted by [L : K].An extension of degree 1 (that is, one where L is equal to K) is called a trivial extension. Extensions of degree 2 and3 are called quadratic extensions and cubic extensions, respectively. Depending on whether the degree is finite orinfinite the extension is called a finite extension or infinite extension.

7.2 Caveats

The notation L /K is purely formal and does not imply the formation of a quotient ring or quotient group or any otherkind of division. Instead the slash expresses the word “over”. In some literature the notation L:K is used.It is often desirable to talk about field extensions in situations where the small field is not actually contained in thelarger one, but is naturally embedded. For this purpose, one abstractly defines a field extension as an injective ringhomomorphism between two fields. Every non-zero ring homomorphism between fields is injective because fields donot possess nontrivial proper ideals, so field extensions are precisely the morphisms in the category of fields.Henceforth, we will suppress the injective homomorphism and assume that we are dealing with actual subfields.

14

7.3. EXAMPLES 15

7.3 Examples

The field of complex numbers C is an extension field of the field of real numbers R, and R in turn is an extensionfield of the field of rational numbersQ. Clearly then, C/Q is also a field extension. We have [C : R] = 2 because {1,i}is a basis, so the extension C/R is finite. This is a simple extension because C=R( i ). [R : Q] = c (the cardinality ofthe continuum), so this extension is infinite.The set Q(√2) = {a + b√2 | a, b ∈ Q} is an extension field of Q, also clearly a simple extension. The degree is 2because {1, √2} can serve as a basis. Q(√2, √3) = Q(√2)( √3)={a + b√3 | a, b ∈ Q(√2)}={a + b√2+ c√3+ d√6 | a,b,c,d ∈ Q} is an extension field of both Q(√2) and Q, of degree 2 and 4 respectively. Finite extensions of Q are alsocalled algebraic number fields and are important in number theory.Another extension field of the rationals, quite different in flavor, is the field of p-adic numbersQp for a prime numberp.It is common to construct an extension field of a given field K as a quotient ring of the polynomial ring K[X] in orderto “create” a root for a given polynomial f(X). Suppose for instance that K does not contain any element x with x2 =−1. Then the polynomial X2 + 1 is irreducible in K[X], consequently the ideal (X2 + 1) generated by this polynomialis maximal, and L = K[X]/(X2 + 1) is an extension field of K which does contain an element whose square is −1(namely the residue class of X).By iterating the above construction, one can construct a splitting field of any polynomial from K[X]. This is anextension field L of K in which the given polynomial splits into a product of linear factors.If p is any prime number and n is a positive integer, we have a finite field GF(pn) with pn elements; this is an extensionfield of the finite field GF(p) = Z/pZ with p elements.Given a field K, we can consider the field K(X) of all rational functions in the variable X with coefficients in K; theelements of K(X) are fractions of two polynomials over K, and indeed K(X) is the field of fractions of the polynomialring K[X]. This field of rational functions is an extension field of K. This extension is infinite.Given a Riemann surface M, the set of all meromorphic functions defined on M is a field, denoted by C(M). It is anextension field of C, if we identify every complex number with the corresponding constant function defined on M.Given an algebraic varietyV over some fieldK, then the function field ofV, consisting of the rational functions definedon V and denoted by K(V), is an extension field of K.

7.4 Elementary properties

If L/K is a field extension, then L and K share the same 0 and the same 1. The additive group (K,+) is a subgroup of(L,+), and the multiplicative group (K−{0},·) is a subgroup of (L−{0},·). In particular, if x is an element of K, thenits additive inverse −x computed in K is the same as the additive inverse of x computed in L; the same is true formultiplicative inverses of non-zero elements of K.In particular then, the characteristics of L and K are the same.

7.5 Algebraic and transcendental elements and extensions

If L is an extension of K, then an element of L which is a root of a nonzero polynomial over K is said to be algebraicover K. Elements that are not algebraic are called transcendental. For example:

• In C/R, i is algebraic because it is a root of x2 + 1.

• In R/Q, √2 + √3 is algebraic, because it is a root[1] of x4 − 10x2 + 1

• In R/Q, e is transcendental because there is no polynomial with rational coefficients that has e as a root (seetranscendental number)

• In C/R, e is algebraic because it is the root of x − e

The special case ofC/Q is especially important, and the names algebraic number and transcendental number are usedto describe the complex numbers that are algebraic and transcendental (respectively) over Q.

16 CHAPTER 7. FIELD EXTENSION

If every element of L is algebraic over K, then the extension L/K is said to be an algebraic extension; otherwise it issaid to be a transcendental extension.A subset S of L is called algebraically independent over K if no non-trivial polynomial relation with coefficients in Kexists among the elements of S. The largest cardinality of an algebraically independent set is called the transcendencedegree of L/K. It is always possible to find a set S, algebraically independent over K, such that L/K(S) is algebraic.Such a set S is called a transcendence basis of L/K. All transcendence bases have the same cardinality, equal to thetranscendence degree of the extension. An extension L/K is said to be purely transcendental if and only if thereexists a transcendence basis S of L/K such that L=K(S). Such an extension has the property that all elements of Lexcept those of K are transcendental over K, but, however, there are extensions with this property which are notpurely transcendental—a class of such extensions take the form L/K where both L and K are algebraically closed.In addition, if L/K is purely transcendental and S is a transcendence basis of the extension, it doesn't necessarilyfollow that L=K(S). (For example, consider the extension Q(x,√x)/Q, where x is transcendental over Q. The set {x}is algebraically independent since x is transcendental. Obviously, the extension Q(x,√x)/Q(x) is algebraic, hence {x}is a transcendence basis. It doesn't generate the whole extension because there is no polynomial expression in x for√x. But it is easy to see that {√x} is a transcendence basis that generates Q(x,√x)), so this extension is indeed purelytranscendental.)It can be shown that an extension is algebraic if and only if it is the union of its finite subextensions. In particular,every finite extension is algebraic. For example,

• C/R and Q(√2)/Q, being finite, are algebraic.

• R/Q is transcendental, although not purely transcendental.

• K(X)/K is purely transcendental.

A simple extension is finite if generated by an algebraic element, and purely transcendental if generated by a tran-scendental element. So

• R/Q is not simple, as it is neither finite nor purely transcendental.

Every field K has an algebraic closure; this is essentially the largest extension field of K which is algebraic over K andwhich contains all roots of all polynomial equations with coefficients in K. For example, C is the algebraic closure ofR.

7.6 Normal, separable and Galois extensions

An algebraic extension L/K is called normal if every irreducible polynomial in K[X] that has a root in L completelyfactors into linear factors over L. Every algebraic extension F/K admits a normal closure L, which is an extension fieldof F such that L/K is normal and which is minimal with this property.An algebraic extension L/K is called separable if the minimal polynomial of every element of L over K is separable,i.e., has no repeated roots in an algebraic closure over K. A Galois extension is a field extension that is both normaland separable.A consequence of the primitive element theorem states that every finite separable extension has a primitive element(i.e. is simple).Given any field extensionL/K, we can consider its automorphismgroupAut(L/K), consisting of all field automorphismsα: L → L with α(x) = x for all x in K. When the extension is Galois this automorphism group is called the Galoisgroup of the extension. Extensions whose Galois group is abelian are called abelian extensions.For a given field extension L/K, one is often interested in the intermediate fields F (subfields of L that contain K).The significance of Galois extensions and Galois groups is that they allow a complete description of the intermediatefields: there is a bijection between the intermediate fields and the subgroups of the Galois group, described by thefundamental theorem of Galois theory.

7.7. GENERALIZATIONS 17

7.7 Generalizations

Field extensions can be generalized to ring extensions which consist of a ring and one of its subrings. A closer non-commutative analog are central simple algebras (CSAs) – ring extensions over a field, which are simple algebra (nonon-trivial 2-sided ideals, just as for a field) and where the center of the ring is exactly the field. For example, the onlyfinite field extension of the real numbers is the complex numbers, while the quaternions are a central simple algebraover the reals, and all CSAs over the reals are Brauer equivalent to the reals or the quaternions. CSAs can be furthergeneralized to Azumaya algebras, where the base field is replaced by a commutative local ring.

7.8 Extension of scalars

Main article: Extension of scalars

Given a field extension, one can "extend scalars" on associated algebraic objects. For example, given a real vectorspace, one can produce a complex vector space via complexification. In addition to vector spaces, one can performextension of scalars for associative algebras defined over the field, such as polynomials or group algebras and theassociated group representations. Extension of scalars of polynomials is often used implicitly, by just considering thecoefficients as being elements of a larger field, but may also be considered more formally. Extension of scalars hasnumerous applications, as discussed in extension of scalars: applications.

7.9 See also• Field theory

• Glossary of field theory

• Tower of fields

• Primary extension

• Regular extension

7.10 Notes[1] “Wolfram|Alpha input: sqrt(2)+sqrt(3)". Retrieved 2010-06-14.

7.11 References• Lang, Serge (2004), Algebra, Graduate Texts in Mathematics 211 (Corrected fourth printing, revised thirded.), New York: Springer-Verlag, ISBN 978-0-387-95385-4

7.12 External links• Hazewinkel, Michiel, ed. (2001), “Extension of a field”, Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4

Chapter 8

Finite field

In mathematics, a finite field or Galois field (so-named in honor of Évariste Galois) is a field that contains a finitenumber of elements. As with any field, a finite field is a set on which the operations of multiplication, addition,subtraction and division are defined and satisfy certain basic rules. The most common examples of finite fields aregiven by the integers mod p when p is a prime number.The number of elements of a finite field is called its order. A finite field of order q exists if and only if the order q isa prime power pk (where p is a prime number and k is a positive integer). All fields of a given order are isomorphic.In a field of order pk, adding p copies of any element always results in zero; that is, the characteristic of the field is p.In a finite field of order q, the polynomial Xq − X has all q elements of the finite field as roots. The non-zero elementsof a finite field form a multiplicative group. This group is cyclic, so all non-zero elements can be expressed as powersof a single element called a primitive element of the field (in general there will be several primitive elements for agiven field.)A field has, by definition, a commutative multiplication operation. A more general algebraic structure that satisfiesall the other axioms of a field but isn't required to have a commutative multiplication is called a division ring (orsometimes skewfield). A finite division ring is a finite field by Wedderburn’s little theorem. This result shows that thefiniteness condition in the definition of a finite field can have algebraic consequences.Finite fields are fundamental in a number of areas of mathematics and computer science, including number theory,algebraic geometry, Galois theory, finite geometry, cryptography and coding theory.

8.1 Definitions, first examples, and basic properties

A finite field is a finite set on which the four operations multiplication, addition, subtraction and division (excludingby zero) are defined, satisfying the rules of arithmetic known as the field axioms. The simplest examples of finitefields are the prime fields: for each prime number p, the field GF(p) (also denoted Z/pZ, Fp , or Fp) of order (that is,size) p is easily constructed as the integers modulo p.The elements of a prime field may be represented by integers in the range 0, ..., p − 1. The sum, the difference andthe product are computed by taking the remainder by p of the integer result. The multiplicative inverse of an elementmay be computed by using the extended Euclidean algorithm (see Extended Euclidean algorithm §Modular integers).Let F be a finite field. For any element x in F and any integer n, let us denote by n⋅x the sum of n copies of x. Theleast positive n such that n⋅1 = 0 must exist and is prime; it is called the characteristic of the field.If the characteristic of F is p, the operation (k, x) 7→ k · xmakes F a GF(p)-vector space. It follows that the numberof elements of F is pn.For every prime number p and every positive integer n, there are finite fields of order pn, and all these fields areisomorphic (see § Existence and uniqueness below). One may therefore identify all fields of order pn, which aretherefore unambiguously denoted Fpn , Fpn or GF(pn), where the letters GF stand for “Galois field”.[1]

The identity

18

8.2. EXISTENCE AND UNIQUENESS 19

(x+ y)p = xp + yp

is true (for every x and y) in a field of characteristic p. (This follows from the fact that all, except the first and the last,binomial coefficients of the expansion of (x+ y)p are multiples of p).For every element x in the prime field GF(p), one has xp = x (This is an immediate consequence of Fermat’s littletheorem, and this may be easily proved as follows: the equality is trivially true for x = 0 and x = 1; one obtains theresult for the other elements of GF(p) by applying the above identity to x and 1, where x successively takes the values1, 2, ..., p − 1 modulo p.) This implies the equality

Xp −X =∏

a∈GF(p)(X − a)

for polynomials over GF(p). More generally, every element in GF(pn) satisfies the polynomial equation xpn − x = 0.Any finite field extension of a finite field is separable and simple. That is, if E is a finite field and F is a subfield ofE, then E is obtained from F by adjoining a single element whose minimal polynomial is separable. To use a jargon,finite fields are perfect.

8.2 Existence and uniqueness

Let q = pn be a prime power, and F be the splitting field of the polynomial

P = Xq −X

over the prime field GF(p). This means that F is a finite field of lowest order, in which P has q distinct roots (theroots are distinct, as the formal derivative of P is equal to −1). Above identity shows that the sum and the product oftwo roots of P are roots of P, as well as the multiplicative inverse of a root of P. In other word, the roots of P form afield of order q, which is equal to F by the minimality of the splitting field.The uniqueness up to isomorphism of splitting fields implies thus that all fields of order q are isomorphic.In summary, we have the following classification theorem first proved in 1893 by E. H. Moore:[2]

The order of a finite field is a prime power. For every prime power q there are fields of orderq, and they are all isomorphic. In these fields, every element satisfies

xq = x,

and the polynomial Xq −X factors as

Xq −X =∏a∈F

(X − a).

It follows that GF(pn) contains a subfield isomorphic to GF(pm) if and only if m is a divisor of n; in that case, thissubfield is unique. In fact, the polynomial Xpm −X divides Xpn −X if and only if m is a divisor of n.

8.3 Explicit construction of finite fields

8.3.1 Non-prime fields

Given a prime power q = pn with p prime and n > 1, the field GF(q) may be explicitly constructed in the followingway. One chooses first an irreducible polynomial P in GF(p)[X] of degree n (such an irreducible polynomial alwaysexists). Then the quotient ring

20 CHAPTER 8. FINITE FIELD

GF(q) = GF(p)[X]/(P )

of the polynomial ring GF(p)[X] by the ideal generated by P is a field of order q.More explicitly, the elements of GF(q) are the polynomials over GF(p) whose degree is strictly less than n. Theaddition and the subtraction are those of polynomials over GF(p). The product of two elements is the remainderof the Euclidean division by P of the product in GF(p)[X]. The multiplicative inverse of a non-zero element maybe computed with the extended Euclidean algorithm; see Extended Euclidean algorithm § Simple algebraic fieldextensions.Except in the construction of GF(4), there are several possible choices for P, which produce isomorphic results. Tosimplify the Euclidean division, for P one commonly chooses polynomials of the form

Xn + aX + b,

which make the needed Euclidean divisions very efficient. However, for some fields, typically in characteristic 2,irreducible polynomials of the formXn + aX + bmay not exist. In characteristic 2, if the polynomial Xn + X + 1 isreducible, it is recommended to choose Xn + Xk + 1 with the lowest possible k that makes the polynomial irreducible.If all these trinomials are reducible, one chooses “pentanomials” Xn + Xa + Xb + Xc + 1, as polynomials of degreegreater than 1, with an even number of terms, are never irreducible in characteristic 2, having 1 as a root.[3]

In the next sections, we will show how this general construction method works for small finite fields.

8.3.2 Field with four elements

Over GF(2), there is only one irreducible polynomial of degree 2:

X2 +X + 1

Therefore, for GF(4) the construction of the preceding section must involve this polynomial, and

GF(4) = GF(2)[X]/(X2 +X + 1).

If one denotes a a root of this polynomial in GF(4), the tables of the operations in GF(4) are the following. Thereis no table for subtraction, as, in every field of characteristic 2, subtraction is identical to addition. In the third table,for the division of x by y, x must be read on the left, and y on the top.

8.3.3 GF(p2) for an odd prime p

For applying the above general construction of finite fields in the case of GF(p2), one has to find an irreduciblepolynomial of degree 2. For p = 2, this has been done in the preceding section. If p is an odd prime, there are alwaysirreducible polynomials of the form X2 − r, with r in GF(p).More precisely, the polynomial X2 − r is irreducible over GF(p) if and only if r is a quadratic non-residue modulop (this is almost the definition of a quadratic non-residue). There are p−1

2 quadratic non-residues modulo p. Forexample, 2 is a quadratic non-residue for p = 3, 5, 11, 13, ..., and 3 is a quadratic non-residue for p = 5, 7, 17, .... Ifp ≡ 3 mod 4, that is p = 3, 7, 11, 19, ..., one may choose −1 ≡ p − 1 as a quadratic non-residue, which allows us tohave a very simple irreducible polynomial X2 + 1.Having chosen a quadratic non-residue r, let α be a symbolic square root of r, that is a symbol which has the propertyα2 = r, in the same way as the complex number i is a symbolic square root of −1. Then, the elements of GF(p2) areall the linear expressions

a+ bα,

8.3. EXPLICIT CONSTRUCTION OF FINITE FIELDS 21

with a and b in GF(p). The operations on GF(p2) are defined as follows (the operations between elements of GF(p)represented by Latin letters are the operations in GF(p)):

−(a+ bα) = −a+ (−b)α

(a+ bα) + (c+ dα) = (a+ c) + (b+ d)α

(a+ bα)(c+ dα) = (ac+ rbd) + (ad+ bc)α

(a+ bα)−1 = a(a2 − rb2)−1 + (−b)(a2 − rb2)−1α

8.3.4 GF(8) and GF(27)

The polynomial

X3 −X − 1

is irreducible over GF(2) and GF(3), that is, it is irreducible modulo 2 and 3 (to show this it suffices to show that it hasno root in GF(2) nor in GF(3)). It follows that the elements of GF(8) and GF(27) may be represented by expressions

a+ bα+ cα2,

where a, b, c are elements of GF(2) or GF(3) (respectively), and α is a symbol such that

α3 = α+ 1.

The addition, additive inverse and multiplication on GF(8) and GF(27) may thus be defined as follows; in followingformulas, the operations between elements of GF(2) or GF(3), represented by Latin letters are the operations in GF(2)or GF(3), respectively:

−(a+ bα+ cα2) = −a+ (−b)α+ (−c)α2 (ForGF (8), identity) the is operation this(a+ bα+ cα2) + (d+ eα+ fα2) = (a+ d) + (b+ e)α+ (c+ f)α2

(a+ bα+ cα2)(d+ eα+ fα2) = (ad+ bf + ce) + (ae+ bd+ bf + ce+ cf)α+ (af + be+ cd+ cf)α2

8.3.5 GF(16)

The polynomial

X4 +X + 1

is irreducible over GF(2), that is, it is irreducible modulo 2. It follows that the elements of GF(16) may be representedby expressions

a+ bα+ cα2 + dα3,

where a, b, c, d are either 0 or 1 (elements of GF(2)), and α is a symbol such that

α4 = α+ 1.

As the characteristic of GF(2) is 2, each element is its additive inverse in GF(16). The addition and multiplicationon GF(16) may be defined as follows; in following formulas, the operations between elements of GF(2), representedby Latin letters are the operations in GF(2).

22 CHAPTER 8. FINITE FIELD

(a+ bα+ cα2 + dα3) + (e+ fα+ gα2 + hα3) = (a+ e) + (b+ f)α+ (c+ g)α2 + (d+ h)α3

(a+ bα+ cα2 + dα3)(e+ fα+ gα2 + hα3) = (ae+ bh+ cg + df) + (af + be+ bh+ cg + df + ch+ dg)α +

(ag + bf + ce+ ch+ dg + dh)α2 + (ah+ bg + cf + de+ dh)α3

8.4 Multiplicative structure

The set of non-zero elements in GF(q) is an abelian group under the multiplication, of order q – 1. By Lagrange’stheorem, there exists a divisor k of q – 1 such that xk = 1 for every non-zero x in GF(q). As the equation Xk = 1 hasat most k solutions in any field, q – 1 is the lowest possible value for k. The structure theorem of finite abelian groupsimplies that this multiplicative group is cyclic, that all non-zero elements are powers of single element. In summary:

The multiplicative group of the non-zero elements in GF(q) is cyclic, and there exist an element a, suchthat the q – 1 non-zero elements of GF(q) are a, a2, ..., aq−2, aq−1 = 1.

Such an element a is called a primitive element. Unless q = 2, 3, the primitive element is not unique. The number ofprimitive elements is φ(q − 1) where φ is Euler’s totient function.Above result implies that xq = x for every x in GF(q). The particular case where q is prime is Fermat’s little theorem.

8.4.1 Discrete logarithm

If a is a primitive element in GF(q), then for any non-zero element x in F, there is a unique integer n with 0 ≤ n ≤ q− 2 such that

x = an.

This integer n is called the discrete logarithm of x to the base a.While the computation of an is rather easy, by using, for example, exponentiation by squaring, the reciprocal oper-ation, the computation of the discrete logarithm is difficult. This has been used in various cryptographic protocols,see Discrete logarithm for details.When the nonzero elements of GF(q) are represented by their discrete logarithms, multiplication and division areeasy, as they reduce to addition and subtraction modulo q – 1. However, addition amounts to computing the discretelogarithm of am + an. The identity

am + an = an(am−n + 1)

allows one to solve this problem by constructing the table of the discrete logarithms of an + 1, called Zech’s logarithms,for n = 0, ..., q − 2 (it is convenient to define the discrete logarithm of zero as being −∞).Zech’s logarithms are useful for large computations, such as linear algebra over medium-sized fields, that is, fields thatare sufficiently large for making natural algorithms inefficient, but not too large, as one has to pre-compute a table ofthe same size as the order of the field.

8.4.2 Roots of unity

Every nonzero element of a finite field is a root of unity, as xq−1 = 1 for every nonzero element of GF(q).If n is a positive integer, an nth primitive root of unity is a solution of the equation xn = 1 that is not a solution ofthe equation xm = 1 for any positive integer m < n. If a is a nth primitive root of unity in a field F, then F contains allthe n roots of unity, which are 1, a, a2, ..., an−1.The field GF(q) contains a nth primitive root of unity if and only if n is a divisor of q − 1; if n is a divisor of q − 1,then the number of primitive nth roots of unity in GF(q) is φ(n) (Euler’s totient function). The number of nth rootsof unity in GF(q) is gcd(n, q − 1).

8.4. MULTIPLICATIVE STRUCTURE 23

In a field of characteristic p, every (np)th root of unity is also a nth root of unity. It follows that primitive (np)th rootsof unity never exist in a field of characteristic p.On the other hand, if n is coprime to p, the roots of the nth cyclotomic polynomial are distinct in every field ofcharacteristic p, as this polynomial is a divisor of Xn − 1, which has 1 as formal derivative. It follows that the nthcyclotomic polynomial factors over GF(p) into distinct irreducible polynomials that have all the same degree, say d,and that GF(pd) is the smallest field of characteristic p that contains the nth primitive roots of unity.

8.4.3 Example

The field GF(64) has several interesting properties that smaller fields do not share. Specifically, it has two subfieldssuch that neither is a subfield of the other, not all generators (elements having a minimal polynomial of degree 6 overGF(2)) are primitive elements, and the primitive elements are not all conjugate under the Galois group.The order of this field being 26, and the divisors of 6 being 1, 2, 3, 6, the subfields of GF(64) are GF(2), GF(22) =GF(4), GF(23) = GF(8), and GF(64) itself. As 2 and 3 are coprime, the intersection of GF(4) and GF(8) in GF(64)is the prime field GF(2).The union of GF(4) and GF(8) has thus 10 elements. The remaining 54 elements of GF(64) generate GF(64) in thesense that no other subfield contains any of them. It follows that they are roots of irreducible polynomials of degree6 over GF(2). This implies that, over GF(2), there are exactly 9 = 54/6 irreducible monic polynomials of degree 6.This may be verified by factoring X64 − X over GF(2).The elements of GF(64) are primitive nth roots of unity for some n dividing 63. As the 3rd and the 7th roots of unitybelong to GF(4) and GF(8), respectively, the 54 generators are primitive nth roots of unity for some n in {9, 21, 63}.Euler’s totient function shows that there are 6 primitive 9th roots of unity, 12 primitive 21st roots of unity, and 36primitive 63rd roots of unity. Summing these numbers, one finds again 54 elements.By factoring the cyclotomic polynomials over GF(2), one finds that:

• The six primitive 9th roots of unity are roots of

X6 +X3 + 1,

and are all conjugate under the action of the Galois group.

• The twelve primitive 21st roots of unity are roots of

(X6 +X4 +X2 +X + 1)(X6 +X5 +X4 +X2 + 1).

They form two orbits under the action of the Galois group. As the two factors are reciprocal to eachother, a root and its (multiplicative) inverse do not belong to the same orbit.

• The 36 primitive elements of GF(64) are the roots of

(X6+X4+X3+X+1)(X6+X+1)(X6+X5+1)(X6+X5+X3+X2+1)(X6+X5+X2+X+1)(X6+X5+X4+X+1),

They split into 6 orbits of 6 elements under the action of the Galois group.

This shows that the best choice to construct GF(64) is to define it as GF(2)[X]/(X6 + X + 1). In fact, this generatoris a primitive element, and this polynomial is the irreducible polynomial that produces the easiest Euclidean division.

24 CHAPTER 8. FINITE FIELD

8.5 Frobenius automorphism and Galois theory

In this section, p is a prime number, and q = pn is a power of p.In GF(q), the identity (x+ y)p = xp + yp implies that the map

φ : x 7→ xp

is a GF(p)-linear endomorphism and a field automorphism of GF(q), which fixes every element of the subfield GF(p).It is called the Frobenius automorphism, after Ferdinand Georg Frobenius.Denoting by φk the composition of φ with itself, k times, we have

φk : x 7→ xpk

.

It has been shown in the preceding section that φn is the identity. For 0 < k < n, the automorphism φk is not theidentity, as, otherwise, the polynomial

Xpk

−X

would have more than pk roots.There are no other GF(p)-automorphisms of GF(q). In other words, GF(pn) has exactly n GF(p)-automorphisms,which are

Id = φ0, φ, φ2, . . . , φn−1.

In terms of Galois theory, this means that GF(pn) is a Galois extension of GF(p), which has a cyclic Galois group.The fact that the Frobenius map is surjective implies that every finite field is perfect.

8.6 Polynomial factorization

Main article: Factorization of polynomials over finite fields

If F is a finite field, a non-constant monic polynomial with coefficients in F is irreducible over F, if it is not the productof two non-constant monic polynomials, with coefficients in F.As every polynomial ring over a field is a unique factorization domain, every monic polynomial over a finite field maybe factored in a unique way (up to the order of the factors) into a product of irreducible monic polynomials.There are efficient algorithms for testing polynomial irreducibility and factoring polynomials over finite field. Theyare a key step for factoring polynomials over the integers or the rational numbers. At least for this reason, everycomputer algebra system has functions for factoring polynomials over finite fields, or, at least, over finite prime fields.

8.6.1 Irreducible polynomials of a given degree

The polynomial

Xq −X

factors into linear factors over a field of order q. More precisely, this polynomial is the product of all monic polyno-mials of degree one over a field of order q.

8.7. APPLICATIONS 25

This implies that, if q = pn that Xq − X is the product of all monic irreducible polynomials over GF(p), whose degreedivides n. In fact, if P is an irreducible factor over GF(p) of Xq − X, its degree divides n, as its splitting field iscontained in GF(pn). Conversely, if P is an irreducible monic polynomial over GF(p) of degree d dividing n, itdefines a field extension of degree d, which is contained in GF(pn), and all roots of P belong to GF(pn), and are rootsof Xq − X; thus P divides Xq − X. As Xq − X does not have any multiple factor, it is thus the product of all theirreducible monic polynomials that divide it.This property is used to compute the product of the irreducible factors of each degree of polynomials over GF(p);see Distinct degree factorization.

8.6.2 Number of monic irreducible polynomials of a given degree over a finite field

The number N(q,n) of monic irreducible polynomials of degree n over GF(q) is given by[4]

N(q, n) =1

n

∑d|n

µ(d)qnd ,

where μ is the Möbius function. This formula is almost a direct consequence of above property of Xq − X.By the above formula, the number of irreducible (not necessarily monic) polynomials of degree n over GF(q) is (q −1)N(q, n).A (slightly simpler) lower bound for N(q, n) is

N(q, n) ≥ 1

n

qn −∑

p|n, pprimeq

np

.

One may easily deduce that, for every q and every n, there is at least one irreducible polynomial of degree n overGF(q). This lower bound is sharp for q = n = 2.

8.7 Applications

In cryptography, the difficulty of the discrete logarithm problem in finite fields or in elliptic curves is the basis ofseveral widely used protocols, such as the Diffie–Hellman protocol. For example, in 2014, the secure connection toWikipedia involves the elliptic curve Diffie–Hellman protocol (ECDHE) over a large finite field.[5] In coding theory,many codes are constructed as subspaces of vector spaces over finite fields.Finite fields are widely used in number theory, as many problems over the integers may be solved by reducing themmodulo one or several prime numbers. For example, the fastest known algorithms for polynomial factorization andlinear algebra over the field of rational numbers proceed by reduction modulo one or several primes, and then recon-struction of the solution by using Chinese remainder theorem, Hensel lifting or the LLL algorithm.Similarly many theoretical problems in number theory can be solved by considering their reductions modulo someor all prime numbers. See, for example, Hasse principle. Many recent developments of algebraic geometry weremotivated by the need to enlarge the power of these modular methods. Wiles’ proof of Fermat’s Last Theorem is anexample of a deep result involving many mathematical tools, including finite fields.

8.8 Extensions

8.8.1 Algebraic closure

A finite field F is not algebraically closed. To demonstrate this, consider the polynomial

f(T ) = 1 +∏α∈F

(T − α),

26 CHAPTER 8. FINITE FIELD

which has no roots in F, since f (α) = 1 for all α in F.The direct limit of the system:

{Fp, Fp2, ..., Fpn, ...},

with inclusion, is an infinite field. It is the algebraic closure of all the fields in the system, and is denoted by: Fp .The inclusions commute with the Frobenius map, as it is defined the same way on each field (x ↦ x p ), so theFrobenius map defines an automorphism of Fp , which carries all subfields back to themselves. In fact Fpn can berecovered as the fixed points of the nth iterate of the Frobenius map.However unlike the case of finite fields, the Frobenius automorphism on Fp has infinite order, and it does not generatethe full group of automorphisms of this field. That is, there are automorphisms of Fp which are not a power of theFrobenius map. However, the group generated by the Frobenius map is a dense subgroup of the automorphism groupin the Krull topology. Algebraically, this corresponds to the additive group Z being dense in the profinite integers(direct product of the p-adic integers over all primes p, with the product topology).If we actually construct our finite fields in such a fashion that Fpn is contained in Fpm whenever n divides m, then thisdirect limit can be constructed as the union of all these fields. Even if we do not construct our fields this way, we canstill speak of the algebraic closure, but some more delicacy is required in its construction.

8.8.2 Wedderburn’s little theorem

A division ring is a generalization of field. Division rings are not assumed to be commutative. There are no non-commutative finite division rings: Wedderburn’s little theorem states that all finite division rings are commutative,hence finite fields. The result holds even if we relax associativity and consider alternative rings, by the Artin–Zorntheorem.

8.8.3 Relationship to other commutative ring classes

Finite fields appear in the following chain of inclusions:

commutative rings ⊃ integral domains ⊃ integrally closed domains ⊃ GCD domains ⊃ uniquefactorization domains ⊃ principal ideal domains ⊃ Euclidean domains ⊃ fields ⊃ finite fields

8.9 See also• Quasi-finite field

• Field with one element

• Finite field arithmetic

• Trigonometry in Galois fields

• Finite ring

• Finite group

• Elementary abelian group

• Hamming space

8.10 Notes[1] This notation was introduced by E. H. Moore in an address given in 1893 at the International Mathematical Congress held

in Chicago Mullen & Panario 2013, p. 10.

8.11. REFERENCES 27

[2] Moore, E. H. (1896), “A doubly-infinite system of simple groups”, in E. H. Moore, et. al., Mathematical Papers Read atthe International Mathematics Congress Held in Connection with the World’s Columbian Exposition, Macmillan & Co., pp.208–242

[3] Recommended Elliptic Curves for Government Use (PDF), National Institute of Standards and Technology, July 1999, p. 3

[4] Jacobson 2009, §4.13

[5] This can be verified by looking at the information on the page provided by the browser.

8.11 References• Jacobson, Nathan (2009) [1985], Basic algebra I (Second ed.), Dover Publications, ISBN 978-0-486-47189-1

• L. Mullen, Garry; Mummert, Carl (2007), Finite Fields and Applications I, Student Mathematical Library(AMS), ISBN 978-0-8218-4418-2

• Mullen, Gary L.; Panario, Daniel (2013), Handbook of Finite Fields, CRC Press, ISBN 978-1-4398-7378-6

• Lidl, Rudolf; Niederreiter, Harald (1997), Finite Fields (2nd ed.), Cambridge University Press, ISBN 0-521-39231-4

8.12 External links• Finite Fields at Wolfram research.

Chapter 9

Galois extension

In mathematics, a Galois extension is an algebraic field extension E/F that is normal and separable; or equivalently,E/F is algebraic, and the field fixed by the automorphism group Aut(E/F) is precisely the base field F. The significanceof being a Galois extension is that the extension has a Galois group and obeys the fundamental theorem of Galoistheory. [1]

A result of Emil Artin allows one to construct Galois extensions as follows: If E is a given field, and G is a finite groupof automorphisms of E with fixed field F, then E/F is a Galois extension.

9.1 Characterization of Galois extensions

An important theorem of Emil Artin states that for a finite extensionE/F, each of the following statements is equivalentto the statement that E/F is Galois:

• E/F is a normal extension and a separable extension.

• E is a splitting field of a separable polynomial with coefficients in F.

• |Aut(E/F)| = [E:F], that is, the number of automorphisms equals the degree of the extension.

Other equivalent statements are:

• Every irreducible polynomial in F[x] with at least one root in E splits over E and is separable.

• |Aut(E/F)| ≥ [E:F], that is, the number of automorphisms is at least the degree of the extension.

• F is the fixed field of a subgroup of Aut(E).

• F is the fixed field of Aut(E/F).

• There is a one-to-one correspondence between subfields of E/F and subgroups of Aut(E/F).

9.2 Examples

There are two basic ways to construct examples of Galois extensions.

• Take any field E, any subgroup of Aut(E), and let F be the fixed field.

• Take any field F, any separable polynomial in F[x], and let E be its splitting field.

Adjoining to the rational number field the square root of 2 gives a Galois extension, while adjoining the cube root of2 gives a non-Galois extension. Both these extensions are separable, because they have characteristic zero. The first

28

9.3. REFERENCES 29

of them is the splitting field of x2 − 2; the second has normal closure that includes the complex cube roots of unity,and so is not a splitting field. In fact, it has no automorphism other than the identity, because it is contained in thereal numbers and x3 − 2 has just one real root. For more detailed examples, see the page on the fundamental theoremof Galois theoryAn algebraic closure K̄ of an arbitrary fieldK is Galois overK if and only ifK is a perfect field.

9.3 References[1] See the article Galois group for definitions of some of these terms and some examples.

9.4 See also• Artin, Emil (1998). Galois Theory. Edited and with a supplemental chapter by Arthur N. Milgram. Mineola,NY: Dover Publications. ISBN 0-486-62342-4. MR 1616156.

• Bewersdorff, Jörg (2006). Galois theory for beginners. Student Mathematical Library 35. Translated fromthe second German (2004) edition by David Kramer. American Mathematical Society. ISBN 0-8218-3817-2.MR 2251389.

• Edwards, HaroldM. (1984). Galois Theory. Graduate Texts in Mathematics 101. New York: Springer-Verlag.ISBN 0-387-90980-X. MR 0743418. (Galois’ original paper, with extensive background and commentary.)

• Funkhouser, H. Gray (1930). “A short account of the history of symmetric functions of roots of equations”.American Mathematical Monthly (The American Mathematical Monthly, Vol. 37, No. 7) 37 (7): 357–365.doi:10.2307/2299273. JSTOR 2299273.

• Hazewinkel, Michiel, ed. (2001), “Galois theory”, Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4

• Jacobson, Nathan (1985). Basic Algebra I (2nd ed.). W.H. Freeman and Company. ISBN 0-7167-1480-9.(Chapter 4 gives an introduction to the field-theoretic approach to Galois theory.)

• Janelidze, G.; Borceux, Francis (2001). Galois theories. Cambridge University Press. ISBN 978-0-521-80309-0. (This book introduces the reader to the Galois theory of Grothendieck, and some generalisations, leadingto Galois groupoids.)

• Lang, Serge (1994). Algebraic Number Theory. Graduate Texts in Mathematics 110 (Second ed.). Berlin,New York: Springer-Verlag. doi:10.1007/978-1-4612-0853-2. ISBN 978-0-387-94225-4. MR 1282723.

• Postnikov, Mikhail Mikhaĭlovich (2004). Foundations of Galois Theory. With a foreword by P. J. Hilton.Reprint of the 1962 edition. Translated from the 1960 Russian original by Ann Swinfen. Dover Publications.ISBN 0-486-43518-0. MR 2043554.

• Rotman, Joseph (1998). Galois Theory (Second ed.). Springer. doi:10.1007/978-1-4612-0617-0. ISBN 0-387-98541-7. MR 1645586.

• Völklein, Helmut (1996). Groups as Galois groups: an introduction. Cambridge Studies in Advanced Mathe-matics 53. Cambridge University Press. doi:10.1017/CBO9780511471117. ISBN 978-0-521-56280-5. MR1405612.

• van der Waerden, Bartel Leendert (1931). Moderne Algebra (in German). Berlin: Springer.. English trans-lation (of 2nd revised edition): Modern algebra. New York: Frederick Ungar. 1949. (Later republished inEnglish by Springer under the title “Algebra”.)

• Pop, Florian (2001). "(Some) New Trends in Galois Theory and Arithmetic” (PDF).

Chapter 10

Irreducible polynomial

This article is about non-factorizable polynomials. For polynomials which are not a composition of polynomials, seepolynomial decomposition.

In mathematics, an irreducible polynomial is, roughly speaking, a non-constant polynomial that cannot be factoredinto the product of two non-constant polynomials. The property of irreducibility depends on the field or ring to whichthe coefficients are considered to belong. For example, the polynomial x2 - 2 is irreducible if the coefficients 1 and−2 are considered as integers, but it factors as (x−

√2)(x+

√2) if the coefficients are considered as real numbers.

One says “the polynomial x2 - 2 is irreducible over the integers but not over the reals”.A polynomial that is not irreducible is sometimes said to be reducible.[1][2] However this term must be used withcare, as it may refer to other notions of reduction.Irreducible polynomials appear naturally in polynomial factorization and algebraic field extensions.It is helpful to compare irreducible polynomials to prime numbers: prime numbers (together with the correspondingnegative numbers of equal magnitude) are the irreducible integers. They exhibit many of the general properties of theconcept of 'irreducibility' that equally apply to irreducible polynomials, such as the essentially unique factorizationinto prime or irreducible factors.

10.1 Definition

If F is a field, a non-constant polynomial is irreducible over F if its coefficients belong to F and it cannot be factoredinto the product of two non-constant polynomials with coefficients in F.A polynomial with integer coefficients, or, more generally, with coefficients in a unique factorization domain R issometimes said to be irreducible over R if it is an irreducible element of the polynomial ring (a polynomial ring overa unique factorization domain is also a unique factorization domain), that is, it is not invertible, nor zero and cannotbe factored into the product of two non-invertible polynomials with coefficients in R. Another definition is frequentlyused, saying that a polynomial is irreducible over R if it is irreducible over the field of fractions of R (the field ofrational numbers, if R is the integers). Both definitions generalize the definition given for the case of coefficients in afield, because, in this case, the non constant polynomials are exactly the polynomials that are non-invertible and nonzero.

10.2 Simple examples

The following six polynomials demonstrate some elementary properties of reducible and irreducible polynomials:

p1(x) = x2 + 4x+ 4 = (x+ 2)(x+ 2)

p2(x) = x2 − 4 = (x− 2)(x+ 2)

30

10.3. OVER THE COMPLEX NUMBERS 31

p3(x) = 9x2 − 3 = 3(3x2 − 1) = 3(x√3− 1)(x

√3 + 1)

p4(x) = x2 − 4

9=

(x− 2

3

)(x+

2

3

)p5(x) = x2 − 2 = (x−

√2)(x+

√2)

p6(x) = x2 + 1 = (x− i)(x+ i)

Over the ring Z of integers, the first three polynomials are reducible (the third one is reducible because the factor3 is not invertible in the integers), the last two are irreducible. (The fourth, of course, is not a polynomial over theintegers.)Over the field Q of rational numbers, the first two and the fourth polynomials are reducible, but the other threepolynomials are irreducible (as a polynomial over the rationals, 3 is a unit, and, therefore, does not count as a factor).Over the field R of real numbers, the first five polynomials are reducible, but p6(x) is still irreducible.Over the field C of complex numbers, all six polynomials are reducible.

10.3 Over the complex numbers

Over the complex field, and, more generally, over an algebraically closed field, a univariate polynomial is irreducibleif and only if its degree is one. This fact is known as the fundamental theorem of algebra in the case of the complexnumbers and, in general, as the condition of being algebraically closed.It follows that every nonconstant univariate polynomial can be factored as

a(x− z1) · · · (x− zn)

where n is the degree, a the leading coefficient and z1, . . . , zn the zeros of the polynomial (not necessarily distinct).There are irreducible multivariate polynomials of every degree over the complex numbers. For example, the poly-nomial

xn + yn − 1,

which defines a Fermat curve, is irreducible for every positive n.

10.4 Over the reals

Over the field of reals, the degree of an irreducible univariate polynomial is either one or two. More precisely, theirreducible polynomials are the polynomials of degree one and the quadratic polynomials ax2 + bx + c that have anegative discriminant b2 − 4ac. It follows that every non-constant univariate polynomial can be factored as a productof polynomials of degree at most two. For example, x4 + 1 factors over the real numbers as (x2 +

√2x+ 1)(x2 −√

2x+ 1), and it cannot be factored further, as both factors have a negative discriminant: (±√2)2 − 4 = −2 < 0.

10.5 Unique factorization property

Main article: Unique factorization domain

Every polynomial over a field F may be factored in a product of a non-zero constant and a finite number of irreducible(over F) polynomials. This decomposition is unique up to the order of the factors and the multiplication of the factorsby non-zero constants whose product is 1.Over a unique factorization domain the same theorem is true, but is more accurately formulated by using the notionof primitive polynomial. A primitive polynomial is a polynomial over a unique factorization domain, such that 1 is agreatest common divisor of its coefficients.

32 CHAPTER 10. IRREDUCIBLE POLYNOMIAL

Let F be a unique factorization domain. A non-constant irreducible polynomial over F is primitive. A primitivepolynomial over F is irreducible over F if and only if it is irreducible over the field of fractions of F. Every polynomialover F may be decomposed into the product of a non zero constant and a finite number of non-constant irreducibleprimitive polynomials. The non-zero constant may itself be decomposed into the product of a unit of F and a finitenumber of irreducible elements ofF. Both factorizations are unique up to the order of the factors and themultiplicationof the factors by a unit of FThis is this theorem which motivates that the definition of irreducible polynomial over a unique factorization domainoften supposes that the polynomial is non-constant.All algorithms which are presently implemented for factoring polynomials over the integers and over the rationalnumbers use this result (see Factorization of polynomials).

10.6 Over the integers

The irreducibility of a polynomial over the integers Z is related to that over the field Fp of p elements (for a prime p). In particular, if a univariate polynomial f over Z is irrreducible over Fp for some prime p that does not divide theleading coefficient of f (the coefficient of the higher power of the variable), then f is irreducible over Z . Eisenstein’scriterion is a variant of this property where irreducibility over p2 is also involved.The converse, however, is not true: there are polynomials of arbitrary large degree that are irreducible over theintegers and reducible over every finite field.[3] A simple example of such a polynomial is x4 + 1.

The relationship between irreducibility over the integers and irreducibility modulo p is deeper than the previous result:to date, all implemented algorithms for factorization and irreducibility over the integers and over the rational numbersuse the factorization over finite fields as a subroutine.

10.7 Algorithms

Main article: Factorization of polynomials

The unique factorization property of polynomials does not mean that the factorization of a given polynomial mayalways be computed. Even the irreducibility of a polynomial may not always be proved by a computation: there arefields over which no algorithm can exist for deciding the irreducibility of arbitrary polynomials.[4]

Algorithms for factoring polynomials and deciding irreducibility are known and implemented in computer algebrasystems for polynomials over the integers, the rational numbers, finite fields and finitely generated field extension ofthese fields. All these algorithms use the algorithms for factorization of polynomials over finite fields.

10.8 Field extension

Main article: Algebraic extension

The notions of irreducible polynomial and of algebraic field extension are strongly related, in the following way.Let x be an element of an extension L of a field K. This element is said to be algebraic if it is a root of a polynomialwith coefficients in K. Among the polynomials of which x is a root, there is exactly one which is monic and of minimaldegree, called the minimal polynomial of x. The minimal polynomial of an algebraic element x of L is irreducible,and is the unique monic irreducible polynomial of which x is a root. The minimal polynomial of x divides everypolynomial which has x as a root (this is Abel’s irreducibility theorem).Conversely, if P (X) ∈ K[X] is a univariate polynomial over a field K, let L = K[X]/P (X) be the quotient ringof the polynomial ring K[X] by the ideal generated by P. Then L is a field if and only if P is irreducible over K. Inthis case, if x is the image of X in L, the minimal polynomial of x is the quotient of P by its leading coefficient.An example of the above is the standard definition of the complex numbers as C = R[X]/(X2 + 1).

If a polynomial P has an irreducible factor Q over K, which has a degree greater than one, one may apply to Q the

10.9. OVER AN INTEGRAL DOMAIN 33

preceding construction of an algebraic extension, for getting an extension in which P has at least one more root thanin K. Iterating this construction, one gets eventually a field over which P factors into linear factors. This field, uniqueup to a field isomorphism, is called the splitting field of P.

10.9 Over an integral domain

If R is an integral domain, an element f of R which is neither zero nor a unit is called irreducible if there are nonon-units g and h with f = gh. One can show that every prime element is irreducible;[5] the converse is not true ingeneral but holds in unique factorization domains. The polynomial ring F[x] over a field F (or any unique-factorizationdomain) is again a unique factorization domain. Inductively, this means that the polynomial ring in n indeterminants(over a ring R) is a unique factorization domain if the same is true for R.

10.10 See also

• Gauss’s lemma (polynomial)

• Rational root theorem, a method of finding whether a polynomial has a linear factor with rational coefficients

• Eisenstein’s criterion

• Perron method

• Hilbert’s irreducibility theorem

• Cohn’s irreducibility criterion

• Irreducible component of a topological space

• Factorization of polynomials over finite fields

• Quartic function#Solving by factoring into quadratics

• Cubic function#Factorization

• Casus irreducibilis, the irreducible cubic with three real roots

• Quadratic equation#Quadratic factorization

10.11 Notes

[1] Gallian 2012, p. 311

[2] Mac Lane and Birkhoff (1999) do not explicitly define “reducible”, but they use it in several places. For example: “For thepresent, we note only that any reducible quadratic or cubic polynomial must have a linear factor.” (p. 268)

[3] David Dummit; Richard Foote (2004). “chapter 9, Proposition 12”. Abtract Algebra. John Wiley & Sons, Inc. p. 309.ISBN 0-471-43334-9.

[4] Fröhlich, A.; Shepherson, J. C. (1955), “On the factorisation of polynomials in a finite number of steps”, MathematischeZeitschrift 62 (1), doi:10.1007/BF01180640, ISSN 0025-5874

[5] Consider p a prime that is reducible: p=ab. Then p | ab => p | a or p | b. Say p | a => a = pc, then we have: p=ab=pcb =>p(1-cb)=0. Because R is a domain we have: cb=1. So b is a unit and p is irreducible

34 CHAPTER 10. IRREDUCIBLE POLYNOMIAL

10.12 References• Lang, Serge (2002), Algebra, Graduate Texts in Mathematics 211 (Revised third ed.), New York: Springer-Verlag, ISBN 978-0-387-95385-4, MR 1878556. This classical book covers most of the content of this article.

• Gallian, Joseph (2012), Contemporary Abstract Algebra (8th ed.), Cengage Learning

• Lidl, Rudolf; Niederreiter, Harald (1997), Finite fields (2nd ed.), Cambridge University Press, ISBN 978-0-521-39231-0, pp. 91.

• Mac Lane, Saunders; Birkhoff, Garrett (1999), Algebra (3rd ed.), American Mathematical Society

• Menezes, Alfred J.; Van Oorschot, Paul C.; Vanstone, Scott A. (1997), Handbook of applied cryptography,CRC Press, ISBN 978-0-8493-8523-0, pp. 154.

10.13 External links• Weisstein, Eric W., “Irreducible Polynomial”, MathWorld.

• Irreducible Polynomial at PlanetMath.org.

• Information on Primitive and Irreducible Polynomials, The (Combinatorial) Object Server.

Chapter 11

Normal extension

In abstract algebra, an algebraic field extension L/K is said to be normal if L is the splitting field of a family ofpolynomials in K[X]. Bourbaki calls such an extension a quasi-Galois extension.

11.1 Equivalent properties and examples

The normality of L/K is equivalent to either of the following properties. LetKa be an algebraic closure ofK containingL.

• Every embedding σ of L in Ka that restricts to the identity on K, satisfies σ(L) = L. In other words, σ is anautomorphism of L over K.

• Every irreducible polynomial in K[X] that has one root in L, has all of its roots in L, that is, it decomposes intolinear factors in L[X]. (One says that the polynomial splits in L.)

If L is a finite extension of K that is separable (for example, this is automatically satisfied if K is finite or has charac-teristic zero) then the following property is also equivalent:

• There exists an irreducible polynomial whose roots, together with the elements of K, generate L. (One says thatL is the splitting field for the polynomial.)

For example, Q(√2) is a normal extension of Q , since it is a splitting field of x2 − 2. On the other hand, Q( 3

√2) is

not a normal extension of Q since the irreducible polynomial x3 − 2 has one root in it (namely, 3√2 ), but not all of

them (it does not have the non-real cubic roots of 2).The fact thatQ( 3

√2) is not a normal extension ofQ can also be seen using the first of the three properties above. The

field A of algebraic numbers is an algebraic closure of Q containing Q( 3√2) . On the other hand,

Q(3√2) = {a+ b

3√2 + c

3√4 ∈ A | a, b, c ∈ Q}

and, if ω is a primitive cubic root of unity, then the map

σ : Q( 3√2) −→ A

a+ b 3√2 + c 3

√4 7→ a+ bω 3

√2 + cω2 3

√4

is an embedding ofQ( 3√2) inA whose restriction toQ is the identity. However, σ is not an automorphism ofQ( 3

√2)

.For any prime p, the extension Q( p

√2, ζp) is normal of degree p(p − 1). It is a splitting field of xp − 2. Here ζp

denotes any pth primitive root of unity. The field Q( 3√2, ζ3) is the normal closure (see below) of Q( 3

√2) .

35

36 CHAPTER 11. NORMAL EXTENSION

11.2 Other properties

Let L be an extension of a field K. Then:

• If L is a normal extension of K and if E is an intermediate extension (i.e., L ⊃ E ⊃ K), then L is a normalextension of E.

• If E and F are normal extensions of K contained in L, then the compositum EF and E ∩ F are also normalextensions of K.

11.3 Normal closure

If K is a field and L is an algebraic extension of K, then there is some algebraic extension M of L such that M is anormal extension of K. Furthermore, up to isomorphism there is only one such extension which is minimal, i.e. suchthat the only subfield ofM which contains L and which is a normal extension of K isM itself. This extension is calledthe normal closure of the extension L of K.If L is a finite extension of K, then its normal closure is also a finite extension.

11.4 See also• Galois extension

• Normal basis

11.5 References• Lang, Serge (2002), Algebra, Graduate Texts in Mathematics 211 (Revised third ed.), New York: Springer-Verlag, ISBN 978-0-387-95385-4, MR 1878556

• Jacobson, Nathan (1989), Basic Algebra II (2nd ed.), W. H. Freeman, ISBN 0-7167-1933-9, MR 1009787

Chapter 12

Quotient ring

In ring theory, a branch of abstract algebra, a quotient ring, also known as factor ring, difference ring[1] or residueclass ring, is a construction quite similar to the factor groups of group theory and the quotient spaces of linearalgebra.[2][3] One starts with a ring R and a two-sided ideal I in R, and constructs a new ring, the quotient ring R/I,whose elements are the cosets of I in R subject to special + and operations.Quotient rings are distinct from the so-called 'quotient field', or field of fractions, of an integral domain as well asfrom the more general 'rings of quotients’ obtained by localization.

12.1 Formal quotient ring construction

Given a ring R and a two-sided ideal I in R, we may define an equivalence relation ~ on R as follows:

a ~ b if and only if a − b is in I.

Using the ideal properties, it is not difficult to check that ~ is a congruence relation. In case a ~ b, we say that a andb are congruent modulo I. The equivalence class of the element a in R is given by

[a] = a + I := { a + r : r in I }.

This equivalence class is also sometimes written as a mod I and called the “residue class of a modulo I".The set of all such equivalence classes is denoted by R/I; it becomes a ring, the factor ring or quotient ring of Rmodulo I, if one defines

• (a + I) + (b + I) = (a + b) + I;

• (a + I)(b + I) = (a b) + I.

(Here one has to check that these definitions are well-defined. Compare coset and quotient group.) The zero-elementof R/I is (0 + I) = I, and the multiplicative identity is (1 + I).The map p from R to R/I defined by p(a) = a + I is a surjective ring homomorphism, sometimes called the naturalquotient map or the canonical homomorphism.

12.2 Examples• The quotient R/{0} is naturally isomorphic to R, and R/R is the zero ring {0}, since, by our definition, for anyr in R , we have that [r]=r +{0}:={r+b : b in {0}} (where {0} is the zero ring), which is isomorphic to R itself. This fits with the general rule of thumb that the larger the ideal I, the smaller the quotient ring R/I. If I is aproper ideal of R, i.e., I ≠ R, then R/I is not the zero ring.

37

38 CHAPTER 12. QUOTIENT RING

• Consider the ring of integers Z and the ideal of even numbers, denoted by 2Z. Then the quotient ring Z/2Zhas only two elements, zero for the even numbers and one for the odd numbers; applying the definition again,[z]=z+2Z:={z+2z: 2z in {2Z}}, where {2Z} is the ideal of even numbers. It is naturally isomorphic to thefinite field with two elements, F2. Intuitively: if you think of all the even numbers as 0, then every integer iseither 0 (if it is even) or 1 (if it is odd and therefore differs from an even number by 1). Modular arithmetic isessentially arithmetic in the quotient ring Z/nZ (which has n elements).

• Now consider the ring R[X] of polynomials in the variable X with real coefficients, and the ideal I = (X2 + 1)consisting of all multiples of the polynomial X2 + 1. The quotient ring R[X]/(X2 + 1) is naturally isomorphicto the field of complex numbers C, with the class [X] playing the role of the imaginary unit i. The reason: we“forced” X2 + 1 = 0, i.e. X2 = −1, which is the defining property of i.

• Generalizing the previous example, quotient rings are often used to construct field extensions. Suppose K issome field and f is an irreducible polynomial in K[X]. Then L = K[X]/(f) is a field whose minimal polynomialover K is f, which contains K as well as an element x = X + (f).

• One important instance of the previous example is the construction of the finite fields. Consider for instancethe field F3 = Z/3Z with three elements. The polynomial f(X) = X2 + 1 is irreducible over F3 (since it hasno root), and we can construct the quotient ring F3[X]/(f). This is a field with 32=9 elements, denoted by F9.The other finite fields can be constructed in a similar fashion.

• The coordinate rings of algebraic varieties are important examples of quotient rings in algebraic geometry. Asa simple case, consider the real variety V = {(x,y) | x2 = y3 } as a subset of the real plane R2. The ring ofreal-valued polynomial functions defined on V can be identified with the quotient ring R[X,Y]/(X2 − Y3), andthis is the coordinate ring of V. The variety V is now investigated by studying its coordinate ring.

• SupposeM is a C∞-manifold, and p is a point ofM. Consider the ring R = C∞(M) of all C∞-functions definedonM and let I be the ideal in R consisting of those functions f which are identically zero in some neighborhoodU of p (where U may depend on f). Then the quotient ring R/I is the ring of germs of C∞-functions on M atp.

• Consider the ring F of finite elements of a hyperreal field *R. It consists of all hyperreal numbers differing froma standard real by an infinitesimal amount, or equivalently: of all hyperreal numbers x for which a standardinteger n with −n < x < n exists. The set I of all infinitesimal numbers in *R, together with 0, is an ideal in F,and the quotient ring F/I is isomorphic to the real numbers R. The isomorphism is induced by associating toevery element x of F the standard part of x, i.e. the unique real number that differs from x by an infinitesimal.In fact, one obtains the same result, namely R, if one starts with the ring F of finite hyperrationals (i.e. ratioof a pair of hyperintegers), see construction of the real numbers.

12.2.1 Alternative complex planes

The quotients R[X]/(X), R[X]/(X + 1), and R[X]/(X − 1) are all isomorphic to R and gain little interest at first.But note that R[X]/(X2) is called the dual number plane in geometric algebra. It consists only of linear binomialsas “remainders” after reducing an element of R[X] by X2. This alternative complex plane arises as a subalgebrawhenever the algebra contains a real line and a nilpotent.Furthermore, the ring quotient R[X]/(X2 − 1) does split into R[X]/(X + 1) and R[X]/(X − 1), so this ring is oftenviewed as the direct sumR⊕R. Nevertheless, an alternative complex number z = x + y j is suggested by j as a root ofX2 − 1, compared to i as root of X2 + 1 = 0. This plane of split-complex numbers normalizes the direct sum R⊕Rby providing a basis {1, j } for 2-space where the identity of the algebra is at unit distance from the zero. With thisbasis a unit hyperbola may be compared to the unit circle of the ordinary complex plane.

12.2.2 Quaternions and alternatives

Suppose X and Y are two, non-commuting, indeterminates and form the free algebra R⟨X,Y ⟩. Then Hamilton’squaternions of 1843 can be cast as

12.3. PROPERTIES 39

R⟨X,Y ⟩/(X2 + 1, Y 2 + 1, XY + Y X).

If Y2 − 1 is substituted for Y2 + 1, then one obtains the ring of split-quaternions. Substituting minus for plus in boththe quadratic binomials also results in split-quaternions. The anti-commutative property YX = −XY implies that XYhas for its square

(XY)(XY) = X(YX)Y = −X(XY)Y = − XXYY = −1.

The three types of biquaternions can also be written as quotients by use of the free algebra with three indeterminatesR⟨X,Y,Z⟩ and constructing appropriate ideals.

12.3 Properties

Clearly, if R is a commutative ring, then so is R/I; the converse however is not true in general.The natural quotient map p has I as its kernel; since the kernel of every ring homomorphism is a two-sided ideal, wecan state that two-sided ideals are precisely the kernels of ring homomorphisms.The intimate relationship between ring homomorphisms, kernels and quotient rings can be summarized as follows:the ring homomorphisms defined on R/I are essentially the same as the ring homomorphisms defined on R that vanish(i.e. are zero) on I. More precisely: given a two-sided ideal I in R and a ring homomorphism f : R→ S whose kernelcontains I, then there exists precisely one ring homomorphism g : R/I → S with gp = f (where p is the natural quotientmap). The map g here is given by the well-defined rule g([a]) = f(a) for all a in R. Indeed, this universal propertycan be used to define quotient rings and their natural quotient maps.As a consequence of the above, one obtains the fundamental statement: every ring homomorphism f : R→ S inducesa ring isomorphism between the quotient ring R/ker(f) and the image im(f). (See also: fundamental theorem onhomomorphisms.)The ideals of R and R/I are closely related: the natural quotient map provides a bijection between the two-sided idealsof R that contain I and the two-sided ideals of R/I (the same is true for left and for right ideals). This relationshipbetween two-sided ideal extends to a relationship between the corresponding quotient rings: ifM is a two-sided idealin R that contains I, and we write M/I for the corresponding ideal in R/I (i.e. M/I = p(M)), the quotient rings R/Mand (R/I)/(M/I) are naturally isomorphic via the (well-defined!) mapping a + M ↦ (a+I) + M/I.In commutative algebra and algebraic geometry, the following statement is often used: If R ≠ {0} is a commutativering and I is a maximal ideal, then the quotient ring R/I is a field; if I is only a prime ideal, then R/I is only an integraldomain. A number of similar statements relate properties of the ideal I to properties of the quotient ring R/I.The Chinese remainder theorem states that, if the ideal I is the intersection (or equivalently, the product) of pairwisecoprime ideals I1,...,Ik, then the quotient ring R/I is isomorphic to the product of the quotient rings R/Ip, p=1,...,k.

12.4 See also

• Residue field

• Goldie’s theorem

12.5 Notes

[1] Jacobson, Nathan (1984). Structure of Rings (revised ed.). American Mathematical Soc. ISBN 0-821-87470-5.

[2] Dummit, David S.; Foote, Richard M. (2004). Abstract Algebra (3rd ed.). John Wiley & Sons. ISBN 0-471-43334-9.

[3] Lang, Serge (2002). Algebra. Graduate Texts in Mathematics. Springer. ISBN 0-387-95385-X.

40 CHAPTER 12. QUOTIENT RING

12.6 Further references• F. Kasch (1978) Moduln und Ringe, translated by DAR Wallace (1982) Modules and Rings, Academic Press,page 33.

• Neal H. McCoy (1948) Rings and Ideals, §13 Residue class rings, page 61, Carus Mathematical Monographs#8, Mathematical Association of America.

• Joseph Rotman (1998). Galois Theory (2nd edition). Springer. pp. 21–3. ISBN 0-387-98541-7.

• B.L. van der Waerden (1970) Algebra, translated by Fred Blum and John R Schulenberger, Frederick UngarPublishing, New York. See Chapter 3.5, “Ideals. Residue Class Rings”, pages 47 to 51.

12.7 External links• Hazewinkel, Michiel, ed. (2001), “Quotient ring”, Encyclopedia ofMathematics, Springer, ISBN978-1-55608-010-4

• Ideals and factor rings from John Beachy’s Abstract Algebra Online

• Quotient ring at PlanetMath.org.

Chapter 13

Root of unity

0

+i

−i

−1 +1

The 5th roots of unity in the complex plane

In mathematics, a root of unity, occasionally called a de Moivre number, is any complex number that gives 1 whenraised to some positive integer power n. Roots of unity are used in many branches of mathematics, and are especiallyimportant in number theory, the theory of group characters, and the discrete Fourier transform.

41

42 CHAPTER 13. ROOT OF UNITY

In field theory and ring theory the notion of root of unity also applies to any ring with a multiplicative identity element.Any algebraically closed field has exactly n nth roots of unity, if n is not divisible by the characteristic of the field.

13.1 General definition

An nth root of unity, where n is a positive integer (i.e. n = 1, 2, 3, …), is a number z satisfying the equation[1][2]

zn = 1.

Traditionally, z is assumed to be a complex number, and subsequent sections of this article will comply with thisusage. Generally, z ∈ R can be considered for any field R, or even for a unital ring. In this general formulation, annth root of unity is just an element of the group of units of order n. Interesting cases are finite fields and modulararithmetics, for which the article root of unity modulo n contains some information.An nth root of unity is primitive if it is not a kth root of unity for some smaller k:

zk ̸= 1 (k = 1, 2, 3, . . . , n− 1).

13.2 Properties

Every nth root of unity z is a primitive ath root of unity for some a where 1 ≤ a ≤ n: if z1 = 1 then z is a primitive firstroot of unity, otherwise if z2 = 1 then z is a primitive second (square) root of unity, otherwise, ..., and by assumptionthere must be a “1” at or before the nth term in the sequence.If z is an nth root of unity and a ≡ b (mod n) then za = zb. By the definition of congruence, a = b + kn for someinteger k. But then,

za = zb+kn = zbzkn = zb(zn)k = zb1k = zb.

Therefore, given a power za of z, it can be assumed that 1 ≤ a ≤ n. This is often convenient.Any integer power of an nth root of unity is also an nth root of unity:

(zk)n = zkn = (zn)k = 1k = 1.

Here k may be negative. In particular, the reciprocal of an nth root of unity is its complex conjugate, and is also annth root of unity:

1

z= z−1 = zn−1 = z̄.

Let z be a primitive nth root of unity. Then the powers z, z2, … , zn −1, zn = z0 = 1 are all distinct. Assume thecontrary, that za = zb where 1 ≤ a < b ≤ n. Then zb − a = 1. But 0 < b − a < n, which contradicts z being primitive.Since an nth-degree polynomial equation can only have n distinct roots, this implies that the powers of a primitiveroot z, z2, … , zn − 1, zn = z0 = 1 are all of the nth roots of unity.From the preceding, it follows that if z is a primitive nth root of unity:

za = zb ⇐⇒ a ≡ b (mod n).

If z is not primitive there is only one implication:

13.3. EXAMPLES 43

a ≡ b (mod n) =⇒ za = zb.

An example showing that the converse implication is false is given by:

n = 4, z = −1, z2 = z4 = 1, 2 ̸≡ 4 (mod 4).

Let z be a primitive nth root of unity and let k be a positive integer. From the above discussion, zk is a primitive athroot of unity for some a. Now if zka = 1, ka must be a multiple of n. The smallest number that is divisible by both nand k is their least common multiple, denoted by lcm(n, k). It is related to their greatest common divisor, gcd(n, k),by the formula:

k n = gcd(k, n) lcm(k, n),

i.e.

lcm(k, n) = kn

gcd(k, n) .

Therefore, zk is a primitive ath root of unity where

a =n

gcd(k, n) .

Thus, if k and n are coprime, zk is also a primitive nth root of unity, and therefore there are φ(n) (where φ is Euler’stotient function) distinct primitive nth roots of unity. (This implies that if n is a prime number, all the roots except+1 are primitive.)In other words, if R(n) is the set of all nth roots of unity and P(n) is the set of primitive ones, R(n) is a disjoint unionof the P(n):

R(n) =∪d |n

P(d),

where the notation means that d goes through all the divisors of n, including 1 and n.Since the cardinality of R(n) is n, and that of P(n) is φ(n), this demonstrates the classical formula

∑d |n

ϕ(d) = n.

13.3 Examples

De Moivre’s formula, which is valid for all real x and integers n, is

(cosx+ i sinx)n = cosnx+ i sinnx.

Setting x = 2π/n gives a primitive nth root of unity:

(cos 2π

n + i sin 2πn

)n= cos 2π + i sin 2π = 1,

44 CHAPTER 13. ROOT OF UNITY

=

The 3rd roots of unity

but for k = 1, 2, ⋯ , n − 1,

(cos 2π

n + i sin 2πn

)k= cos 2kπ

n + i sin 2kπn ̸= 1

This formula shows that on the complex plane the nth roots of unity are at the vertices of a regular n-sided polygoninscribed in the unit circle, with one vertex at 1. (See the plots for n = 3 and n = 5 on the right.) This geometricfact accounts for the term “cyclotomic” in such phrases as cyclotomic field and cyclotomic polynomial; it is from theGreek roots "cyclo" (circle) plus "tomos" (cut, divide).Euler’s formula

eix = cosx+ i sinx,

which is valid for all real x, can be used to put the formula for the nth roots of unity into the form

e2πikn 0 ≤ k < n.

It follows from the discussion in the previous section that this is a primitive nth-root if and only if the fraction k/n isin lowest terms, i.e. that k and n are coprime.The roots of unity are trigonometric numbers and are, by definition, the roots of a polynomial equation and are thusalgebraic numbers. In fact, Galois theory can be used to show that they may be expressed as expressions involvingintegers and the operations of addition, subtraction, multiplication, division, and the extraction of roots. (There aremore details later in this article at Cyclotomic fields.)

13.3. EXAMPLES 45

Plot of z3 − 1, in which a zero is represented by the color black.

The equation z1 = 1 obviously has only one solution, +1, which is therefore the only primitive first root of unity. It isa nonprimitive 2nd, 3rd, 4th, ... root of unity.The equation z2 = 1 has two solutions, +1 and −1. +1 is the primitive first root of unity, leaving −1 as the onlyprimitive second (square) root of unity. It is a nonprimitive 4th, 6th, 8th, ...root of unity.The only real roots of unity are ±1; all the others are non-real complex numbers, as can be seen from de Moivre’sformula or the figures.The third (cube) roots satisfy the equation z3 − 1 = 0; the non-principal root +1 may be factored out, giving (z − 1)(z2+ z + 1) = 0. Therefore, the primitive cube roots of unity are the roots of a quadratic equation. (See Cyclotomicpolynomial, below.)

{e

2πi3 , e−

2πi3

}=

{−1 + i

√3

2,−1− i

√3

2

}The two primitive fourth roots of unity are the two square roots of the primitive square root of unity, −1

{e

2πi4 , e−

2πi4

}={±√−1}= {+i,−i} .

46 CHAPTER 13. ROOT OF UNITY

Plot of z5 − 1, in which a zero is represented by the color black.

The four primitive fifth roots of unity are

{e

2πik5

∣∣∣ 1 ≤ k ≤ 4}=

u√5− 1

4+ v i

√5 + u

√5

8

∣∣∣∣∣∣u, v ∈ {−1, 1}

.

The two primitive sixth roots of unity are the negatives (and also the square roots) of the two primitive cube roots:

{e

2πi6 , e−

2πi6

}=

{1 + i

√3

2,1− i

√3

2

}.

Gauss observed that if a primitive nth root of unity can be expressed using only square roots, then it is possible toconstruct the regular n-gon using only ruler and compass, and that if the root of unity requires third or fourth orhigher radicals the regular polygon cannot be constructed. The 7th roots of unity are the first that require cube roots.Note that the real part and imaginary part are both real numbers, but complex numbers are buried in the expressions.They cannot be removed. See casus irreducibilis for details.One of the primitive seventh roots of unity is

13.4. PERIODICITY 47

e2πi7 =

−1 + 3

√7+21

√−3

2 + 3

√7−21

√−3

2

6+

i

2

√√√√7− ω2 3

√7+21

√−3

2 − ω 3

√7−21

√−3

2

3

where ω and ω2 are the primitive cube roots of unity exp(2πi/3) and exp(4πi/3).The four primitive eighth roots of unity are ± the square roots of the primitive fourth roots, ±i. One of them is:

e2πi8 =

√i =

√2

2+ i

√2

2.

See heptadecagon for the real part of a 17th root of unity.

13.4 Periodicity

If z is a primitive nth root of unity, then the sequence of powers

… , z−1, z0, z1, …

is n-periodic (because z j + n = z j ⋅z n = z j ⋅1 = z j for all values of j), and the n sequences of powers

sk: … , z k⋅(−1), z k⋅0, z k⋅1, …

for k = 1, … , n are all n-periodic (because z k⋅(j + n) = z k⋅j). Furthermore, the set {s1, … , sn} of these sequences is abasis of the linear space of all n-periodic sequences. This means that any n-periodic sequence of complex numbers

… , x₋₁ , x0 , x1, …

can be expressed as a linear combination of powers of a primitive nth root of unity:

xj =∑k

Xk · zk·j = X1z1·j + · · ·+Xn · zn·j

for some complex numbers X1, … , Xn and every integer j.This is a form of Fourier analysis. If j is a (discrete) time variable, then k is a frequency and Xk is a complexamplitude.Choosing for the primitive nth root of unity

z = e2πi/n = cos(2π/n) + i⋅sin(2π/n)

allows xj to be expressed as a linear combination of cos and sin:

xj = ∑k Ak⋅cos(2π⋅j⋅k/n) + ∑k Bk⋅sin(2π⋅j⋅k/n).

This is a discrete Fourier transform.

13.5 Summation

Let SR(n) be the sum of all the nth roots of unity, primitive or not. Then

48 CHAPTER 13. ROOT OF UNITY

SR(n) ={1, n = 1

0, n > 1.

For n = 1 there is nothing to prove. For n > 1, it is “intuitively obvious” from the symmetry of the roots in the complexplane. For a rigorous proof, let z be a primitive nth root of unity. Then the set of all roots is given by zk, k = 0, 1, …, n − 1, and their sum is given by the formula for a geometric series:

n−1∑k=0

zk =zn − 1

z − 1= 0.

Let SP(n) be the sum of all the primitive nth roots of unity. Then

SP(n) = µ(n),

where μ(n) is the Möbius function.In the section Elementary facts, it was shown that if R(n) is the set of all nth roots of unity and P(n) is the set ofprimitive ones, R(n) is a disjoint union of the P(n):

R(n) =∪d |n

P(d),

This implies

SR(n) =∑d |n

SP(d).

Applying the Möbius inversion formula gives

SP(n) =∑d |n

µ(d) SR(nd

).

In this formula, if d < n, then SR(n/d) = 0, and for d = n: SR(n/d) = 1. Therefore, SP(n) = μ(n).This is the special case cn(1) of Ramanujan’s sum cn(s), defined as the sum of the sth powers of the primitive nthroots of unity:

cn(s) =n∑

a=1gcd(a,n)=1

e2πians.

13.6 Orthogonality

From the summation formula follows an orthogonality relationship: for j = 1, … , n and j′ = 1, … , n

n∑k=1

zj·k · zj′·k = n · δj,j′

where δ is the Kronecker delta and z is any primitive nth root of unity.

13.7. CYCLOTOMIC POLYNOMIALS 49

The n × n matrix U whose (j, k)th entry is

Uj,k = n− 12 · zj·k

defines a discrete Fourier transform. Computing the inverse transformation using gaussian elimination requires O(n3)operations. However, it follows from the orthogonality that U is unitary. That is,

n∑k=1

Uj,k · Uk,j′ = δj,j′ ,

and thus the inverse of U is simply the complex conjugate. (This fact was first noted by Gauss when solving theproblem of trigonometric interpolation). The straightforward application of U or its inverse to a given vector requiresO(n2) operations. The fast Fourier transform algorithms reduces the number of operations further to O(n log n).

13.7 Cyclotomic polynomials

Main article: Cyclotomic polynomial

The zeroes of the polynomial

p(z) = zn − 1

are precisely the nth roots of unity, each with multiplicity 1. The nth cyclotomic polynomial is defined by the factthat its zeros are precisely the primitive nth roots of unity, each with multiplicity 1.

Φn(z) =

φ(n)∏k=1

(z − zk)

where z1, z2, z3, … ,zᵩ₍n₎ are the primitive nth roots of unity, and φ(n) is Euler’s totient function. The polynomialΦn(z) has integer coefficients and is an irreducible polynomial over the rational numbers (i.e., it cannot be written asthe product of two positive-degree polynomials with rational coefficients). The case of prime n, which is easier thanthe general assertion, follows by applying Eisenstein’s criterion to the polynomial

(z + 1)n − 1

((z + 1)− 1),

and expanding via the binomial theorem.Every nth root of unity is a primitive dth root of unity for exactly one positive divisor d of n. This implies that

zn − 1 =∏d |n

Φd(z).

This formula represents the factorization of the polynomial zn − 1 into irreducible factors.

z1 − 1 = z − 1z2 − 1 = (z − 1)⋅(z + 1)z3 − 1 = (z − 1)⋅(z2 + z + 1)z4 − 1 = (z − 1)⋅(z + 1)⋅(z2 + 1)z5 − 1 = (z − 1)⋅(z4 + z3 + z2 + z + 1)z6 − 1 = (z − 1)⋅(z + 1)⋅(z2 + z + 1)⋅(z2 − z + 1)z7 − 1 = (z − 1)⋅(z6 + z5 + z4 + z3 + z2 +z + 1)

50 CHAPTER 13. ROOT OF UNITY

Applying Möbius inversion to the formula gives

Φn(z) =∏d |n

(zn/d − 1)µ(d) =∏d |n

(zd − 1)µ(n/d),

where μ is the Möbius function.So the first few cyclotomic polynomials are

Φ1(z) = z − 1Φ2(z) = (z2 − 1)⋅(z − 1)−1 = z + 1Φ3(z) = (z3 − 1)⋅(z − 1)−1 = z2 + z + 1Φ4(z) = (z4 − 1)⋅(z2 − 1)−1 = z2 + 1Φ5(z) = (z5 − 1)⋅(z − 1)−1 = z4 + z3 + z2 + z + 1Φ6(z) = (z6 − 1)⋅(z3 − 1)−1⋅(z2 − 1)−1⋅(z − 1) = z2 − z + 1Φ7(z) = (z7 − 1)⋅(z − 1)−1 = z6 + z5 + z4 + z3 + z2 +z + 1.

If p is a prime number, then all the pth roots of unity except 1 are primitive pth roots, and we have

Φp(z) =zp − 1

z − 1=

p−1∑k=0

zk.

Substituting any positive integer ≥ 2 for z, this sum becomes a base z repunit. Thus a necessary (but not sufficient)condition for a repunit to be prime is that its length be prime.Note that, contrary to first appearances, not all coefficients of all cyclotomic polynomials are 0, 1, or −1. The firstexception is Φ₁₀₅. It is not a surprise it takes this long to get an example, because the behavior of the coefficientsdepends not so much on n as on how many odd prime factors appear in n. More precisely, it can be shown that if nhas 1 or 2 odd prime factors (e.g., n = 150) then the nth cyclotomic polynomial only has coefficients 0, 1 or −1. Thusthe first conceivable n for which there could be a coefficient besides 0, 1, or −1 is a product of the three smallest oddprimes, and that is 3⋅5⋅7 = 105. This by itself doesn't prove the 105th polynomial has another coefficient, but doesshow it is the first one which even has a chance of working (and then a computation of the coefficients shows it does).A theorem of Schur says that there are cyclotomic polynomials with coefficients arbitrarily large in absolute value. Inparticular, if n = p1⋅p2⋅ ⋯ ⋅pt, where p1 < p2 < ⋯ < pt are odd primes, p1 + p2 > pt, and t is odd, then 1 − t occursas a coefficient in the nth cyclotomic polynomial.[3]

Many restrictions are known about the values that cyclotomic polynomials can assume at integer values. For example,if p is prime and d ∣ Φp(d), then either d ≡ 1 (mod p), or d ≡ 0 (mod p).Cyclotomic polynomials are solvable in radicals, as roots of unity are themselves radicals. Moreover, there exist moreinformative radical expressions for nth roots of unity with the additional property[4] that every value of the expressionobtained by choosing values of the radicals (for example, signs of square roots) is a primitive nth root of unity. Thiswas already shown by Gauss in 1797.[5] Efficient algorithms exist for calculating such expressions.[6]

13.8 Cyclic groups

The nth roots of unity form under multiplication a cyclic group of order n, and in fact these groups comprise all ofthe finite subgroups of the multiplicative group of the complex number field. A generator for this cyclic group is aprimitive nth root of unity.The nth roots of unity form an irreducible representation of any cyclic group of order n. The orthogonality relationshipalso follows from group-theoretic principles as described in character group.The roots of unity appear as entries of the eigenvectors of any circulant matrix, i.e. matrices that are invariantunder cyclic shifts, a fact that also follows from group representation theory as a variant of Bloch’s theorem.[7] Inparticular, if a circulant Hermitian matrix is considered (for example, a discretized one-dimensional Laplacian withperiodic boundaries[8]), the orthogonality property immediately follows from the usual orthogonality of eigenvectorsof Hermitian matrices.

13.9. CYCLOTOMIC FIELDS 51

13.9 Cyclotomic fields

Main article: Cyclotomic field

By adjoining a primitive nth root of unity to Q, one obtains the nth cyclotomic field Q(exp(2πi/n)). This field con-tains all nth roots of unity and is the splitting field of the nth cyclotomic polynomial over Q. The field extensionQ(exp(2πi/n))/Q has degree φ(n) and its Galois group is naturally isomorphic to the multiplicative group of units ofthe ring Z/nZ.As the Galois group of Q(exp(2πi/n))/Q is abelian, this is an abelian extension. Every subfield of a cyclotomic fieldis an abelian extension of the rationals. It follows that every nth root of unity may be expressed in term of k-roots,with various k not exceeding φ(n). In these cases Galois theory can be written out explicitly in terms of Gaussianperiods: this theory from the Disquisitiones Arithmeticae of Gauss was published many years before Galois.[9]

Conversely, every abelian extension of the rationals is such a subfield of a cyclotomic field – this is the content of atheorem of Kronecker, usually called the Kronecker–Weber theorem on the grounds that Weber completed the proof.

13.10 Relation to quadratic integers

0 1 2ϕ−1−ϕ

In the complex plane, the red points are the fifth roots of unity, and the blue points are the sums of a fifth root of unit and its complexconjugate.

For n = 2, both roots of unity 1 and −1 belong to Z.For three values of n, the roots of unity are quadratic integers:

• For n = 3, 6 they are Eisenstein integers (D = −3).

• For n = 4 they are Gaussian integers (D = −1): see imaginary unit.

For four other values of n, the primitive roots of unity are not quadratic integers, but the some of any root of unitywith its complex conjugate (also a nth root of unity) is a quadratic integer.For n = 5, 10, neither of non-real roots of unity (which satisfy a quartic equation) is a quadratic integer, but the sumz + z = 2 Rez of each root with its complex conjugate (also a 5th root of unity) is an element of the ring Z[1 + √5/2](D = 5). For two pairs of non-real 5th roots of unity these sums are inverse golden ratio and minus golden ratio.For n = 8, for any root of unity z + z equals to either ±2, 0, or ±√2 (D = 2).For n = 12, for any root of unity, z + z equals to either 0, ±1, ±2 or ±√3 (D = 3).

52 CHAPTER 13. ROOT OF UNITY

In the complex plane, the corners of the two squares are the eighth roots of unity

13.11 See also• Argand system

• Circle group, the unit complex numbers

• Group scheme of roots of unity

• Primitive root modulo n

• Dirichlet character

• Ramanujan’s sum

13.12 Notes[1] Hadlock, Charles R. (2000). Field Theory and Its Classical Problems, Volume 14. Cambridge University Press. pp. 84–86.

ISBN 978-0-88385-032-9.

[2] Lang, Serge (2002). “Roots of unity”. Algebra. Springer. pp. 276–277. ISBN 978-0-387-95385-4.

13.13. REFERENCES 53

[3] Emma Lehmer, On the magnitude of the coefficients of the cyclotomic polynomial, Bulletin of the American MathematicalSociety 42 (1936), no. 6, pp. 389–392.

[4] Landau, Susan; Miller, Gary L. (1985). “Solvability by radicals is in polynomial time”. Journal of Computer and SystemSciences 30 (2): 179–208. doi:10.1016/0022-0000(85)90013-3.

[5] Gauss, Carl F. (1965). Disquisitiones Arithmeticae. Yale University Press. pp. §§359–360. ISBN 0-300-09473-6.

[6] Weber, Andreas; Keckeisen, Michael. “Solving Cyclotomic Polynomials by Radical Expressions” (PDF). Retrieved 2007-06-22.

[7] T. Inui, Y. Tanabe, and Y. Onodera, Group Theory and Its Applications in Physics (Springer, 1996).

[8] Gilbert Strang, "The discrete cosine transform,” SIAM Review 41 (1), 135–147 (1999).

[9] The Disquisitiones was published in 1801, Galois was born in 1811, died in 1832, but wasn't published until 1846.

13.13 References• Lang, Serge (2002), Algebra, Graduate Texts in Mathematics 211 (Revised third ed.), New York: Springer-Verlag, ISBN 978-0-387-95385-4, Zbl 0984.00001, MR 1878556

• Milne, James S. (1998). “Algebraic Number Theory”. Course Notes.

• Milne, James S. (1997). “Class Field Theory”. Course Notes.

• Neukirch, Jürgen (1999), Algebraic Number Theory, Grundlehren der mathematischen Wissenschaften 322,Berlin: Springer-Verlag, ISBN 978-3-540-65399-8, Zbl 0956.11021, MR 1697859

• Neukirch, Jürgen (1986). Class Field Theory. Berlin: Springer-Verlag. ISBN 3-540-15251-2.

• Washington, Lawrence C. (1997). Cyclotomic fields (2nd ed.). New York: Springer-Verlag. ISBN 0-387-94762-0.

• Derbyshire, John (2006). “Roots of Unity”. Unknown Quantity. Washington, D.C.: Joseph Henry Press.ISBN 0-309-09657-X.

13.14 Further reading• Storer, Thomas (1967). Cyclotomy and difference sets. Chicago: MarkhamPublishingCompany. Zbl 0157.03301.

Chapter 14

Separable extension

In the subfield of algebra named field theory, a separable extension is an algebraic field extension E ⊃ F such thatfor every α ∈ E , the minimal polynomial of α over F is a separable polynomial (i.e., has distinct roots; see below forthe definition in this context).[1] Otherwise, the extension is called inseparable. There are other equivalent definitionsof the notion of a separable algebraic extension, and these are outlined later in the article.The importance of separable extensions lies in the fundamental role they play in Galois theory in finite characteristic.More specifically, a finite degree field extension is Galois if and only if it is both normal and separable.[2] Sincealgebraic extensions of fields of characteristic zero, and of finite fields, are separable, separability is not an obstaclein most applications of Galois theory.[3][4] For instance, every algebraic (in particular, finite degree) extension of thefield of rational numbers is necessarily separable.Despite the ubiquity of the class of separable extensions in mathematics, its extreme opposite, namely the class ofpurely inseparable extensions, also occurs quite naturally. An algebraic extension E ⊃ F is a purely inseparableextension if and only if for every α ∈ E \F , the minimal polynomial of α over F is not a separable polynomial (i.e.,does not have distinct roots).[5] For a field F to possess a non-trivial purely inseparable extension, it must necessarilybe an infinite field of prime characteristic (i.e. specifically, imperfect), since any algebraic extension of a perfect fieldis necessarily separable.[3]

14.1 Informal discussion

An arbitrary polynomial f with coefficients in some field F is said to have distinct roots if and only if it has deg(f)roots in some extension field E ⊇ F . For instance, the polynomial g(X)=X2+1 with real coefficients has preciselydeg(g)=2 roots in the complex plane; namely the imaginary unit i, and its additive inverse −i, and hence does havedistinct roots. On the other hand, the polynomial h(X)=(X−2)2 with real coefficients does not have distinct roots; only2 can be a root of this polynomial in the complex plane and hence it has only one, and not deg(h)=2 roots.To test if a polynomial has distinct roots, it is not necessary to consider explicitly any field extension nor to compute theroots: a polynomial has distinct roots if and only if the greatest common divisor of the polynomial and its derivativeis a constant. For instance, the polynomial g(X)=X2+1 in the above paragraph, has 2X as derivative, and, over a fieldof characteristic different from 2, we have g(X) - (1/2 X) 2X = 1, which proves, by Bézout’s identity, that the greatestcommon divisor is a constant. On the other hand, over a field where 2=0, the greatest common divisor is g, and wehave g(X) = (X+1)2 has 1=−1 as double root. On the other hand, the polynomial h does not have distinct roots,whichever is the field of the coefficients, and indeed, h(X)=(X−2)2, its derivative is 2 (X−2) and divides it, and hencedoes have a factor of the form (X − α)2 for α = 2 ).Although an arbitrary polynomial with rational or real coefficients may not have distinct roots, it is natural to ask atthis stage whether or not there exists an irreducible polynomial with rational or real coefficients that does not havedistinct roots. The polynomial h(X)=(X−2)2 does not have distinct roots but it is not irreducible as it has a non-trivialfactor (X−2). In fact, it is true that there is no irreducible polynomial with rational or real coefficients that does nothave distinct roots; in the language of field theory, every algebraic extension of Q or R is separable and hence bothof these fields are perfect.

54

14.2. SEPARABLE AND INSEPARABLE POLYNOMIALS 55

14.2 Separable and inseparable polynomials

A polynomial f in F[X] is a separable polynomial if and only if every irreducible factor of f in F[X] has distinctroots.[6] The separability of a polynomial depends on the field in which its coefficients are considered to lie; forinstance, if g is an inseparable polynomial in F[X], and one considers a splitting field, E, for g over F, g is necessarilyseparable in E[X] since an arbitrary irreducible factor of g in E[X] is linear and hence has distinct roots.[1] Despitethis, a separable polynomial h in F[X] must necessarily be separable over every extension field of F.[7]

Let f in F[X] be an irreducible polynomial and f ' its formal derivative. Then the following are equivalent conditionsfor f to be separable; that is, to have distinct roots:

• If E ⊇ F and α ∈ E , then (X − α)2 does not divide f in E[X].[8]

• There existsK ⊇ F such that f has deg(f) roots in K.[8]

• f and f ' do not have a common root in any extension field of F.[9]

• f ' is not the zero polynomial.[10]

By the last condition above, if an irreducible polynomial does not have distinct roots, its derivative must be zero.Since the formal derivative of a positive degree polynomial can be zero only if the field has prime characteristic,for an irreducible polynomial to not have distinct roots its coefficients must lie in a field of prime characteristic.More generally, if an irreducible (non-zero) polynomial f in F[X] does not have distinct roots, not only must thecharacteristic of F be a (non-zero) prime number p, but also f(X)=g(Xp) for some irreducible polynomial g in F[X].[11]By repeated application of this property, it follows that in fact, f(X) = g(Xpn

) for a non-negative integer n andsome separable irreducible polynomial g in F[X] (where F is assumed to have prime characteristic p).[12]

By the property noted in the above paragraph, if f is an irreducible (non-zero) polynomial with coefficients in thefield F of prime characteristic p, and does not have distinct roots, it is possible to write f(X)=g(Xp). Furthermore,if g(X) =

∑aiX

i , and if the Frobenius endomorphism of F is an automorphism, g may be written as g(X) =∑bpiX

i , and in particular, f(X) = g(Xp) =∑

bpiXpi = (

∑biX

i)p ; a contradiction of the irreducibility of f.Therefore, if F[X] possesses an inseparable irreducible (non-zero) polynomial, then the Frobenius endomorphism ofF cannot be an automorphism (where F is assumed to have prime characteristic p).[13]

If K is a finite field of prime characteristic p, and if X is an indeterminant, then the field of rational functions overK, K(X), is necessarily imperfect. Furthermore, the polynomial f(Y)=Yp−X is inseparable.[1] (To see this, note thatthere is some extension fieldE ⊇ K(X) in which f has a root α ; necessarily, αp = X in E. Therefore, working overE, f(Y ) = Y p −X = Y p − αp = (Y − α)p (the final equality in the sequence follows from freshman’s dream),and f does not have distinct roots.) More generally, if F is any field of (non-zero) prime characteristic for which theFrobenius endomorphism is not an automorphism, F possesses an inseparable algebraic extension.[14]

A field F is perfect if and only if all of its algebraic extensions are separable (in fact, all algebraic extensions of Fare separable if and only if all finite degree extensions of F are separable). By the argument outlined in the aboveparagraphs, it follows that F is perfect if and only if F has characteristic zero, or F has (non-zero) prime characteristicp and the Frobenius endomorphism of F is an automorphism.

14.3 Properties

• IfE ⊇ F is an algebraic field extension, and if α, β ∈ E are separable over F, then α+β and αβ are separableover F. In particular, the set of all elements in E separable over F forms a field.[15]

• If E ⊇ L ⊇ F is such that E ⊇ L and L ⊇ F are separable extensions, then E ⊇ F is separable.[16]Conversely, if E ⊇ F is a separable algebraic extension, and if L is any intermediate field, then E ⊇ L andL ⊇ F are separable extensions.[17]

• If E ⊇ F is a finite degree separable extension, then it has a primitive element; i.e., there exists α ∈ E withE = F [α] . This fact is also known as the primitive element theorem or Artin’s theorem on primitive elements.

56 CHAPTER 14. SEPARABLE EXTENSION

14.4 Separable extensions within algebraic extensions

Separable extensions occur quite naturally within arbitrary algebraic field extensions. More specifically, if E ⊇ Fis an algebraic extension and if S = {α ∈ E|α is separable over F} , then S is the unique intermediate field thatis separable over F and over which E is purely inseparable.[18] If E ⊇ F is a finite degree extension, the degree [S: F] is referred to as the separable part of the degree of the extension E ⊇ F (or the separable degree of E/F),and is often denoted by [E : F] ₑ or [E : F] .[19] The inseparable degree of E/F is the quotient of the degree bythe separable degree. When the characteristic of F is p > 0, it is a power of p.[20] Since the extension E ⊇ F isseparable if and only if S = E , it follows that for separable extensions, [E : F]=[E : F] ₑ , and conversely. If E ⊇ Fis not separable (i.e., inseparable), then [E : F] ₑ is necessarily a non-trivial divisor of [E : F], and the quotient isnecessarily a power of the characteristic of F.[19]

On the other hand, an arbitrary algebraic extensionE ⊇ F may not possess an intermediate extension K that is purelyinseparable over F and over which E is separable (however, such an intermediate extension does exist when E ⊇ Fis a finite degree normal extension (in this case, K can be the fixed field of the Galois group of E over F)). If suchan intermediate extension does exist, and if [E : F] is finite, then if S is defined as in the previous paragraph, [E :F] ₑ =[S : F]=[E : K].[21] One known proof of this result depends on the primitive element theorem, but there doesexist a proof of this result independent of the primitive element theorem (both proofs use the fact that if K ⊇ F isa purely inseparable extension, and if f in F[X] is a separable irreducible polynomial, then f remains irreducible inK[X][22]). The equality above ([E : F] ₑ =[S : F]=[E : K]) may be used to prove that if E ⊇ U ⊇ F is such that [E :F] is finite, then [E : F] ₑ =[E : U] ₑ [U : F] ₑ .[23]

If F is any field, the separable closure Fsep of F is the field of all elements in an algebraic closure of F that areseparable over F. This is the maximal Galois extension of F. By definition, F is perfect if and only if its separable andalgebraic closures coincide (in particular, the notion of a separable closure is only interesting for imperfect fields).

14.5 The definition of separable non-algebraic extension fields

Although many important applications of the theory of separable extensions stem from the context of algebraic fieldextensions, there are important instances in mathematics where it is profitable to study (not necessarily algebraic)separable field extensions.Let F/k be a field extension and let p be the characteristic exponent of k .[24] For any field extension L of k, we writeFL = L ⊗k F (cf. Tensor product of fields.) Then F is said to be separable over k if the following equivalentconditions are met:

• F p and k are linearly disjoint over kp

• Fk1/p is reduced.

• FL is reduced for all field extensions L of k.

(In other words, F is separable over k if F is a separable k-algebra.)A separating transcendence basis for F/k is an algebraically independent subset T of F such that F/k(T) is a finiteseparable extension. An extension E/k is separable if and only if every finitely generated subextension F/k of E/k hasa separating transcendence basis.[25]

Suppose there is some field extension L of k such that FL is a domain. Then F is separable over k if and only if thefield of fractions of FL is separable over L.An algebraic element of F is said to be separable over k if its minimal polynomial is separable. If F/k is an algebraicextension, then the following are equivalent.

• F is separable over k.

• F consists of elements that are separable over k.

• Every subextension of F/k is separable.

• Every finite subextension of F/k is separable.

14.6. DIFFERENTIAL CRITERIA 57

If F/k is finite extension, then the following are equivalent.

• (i) F is separable over k.

• (ii) F = k(a1, ..., ar) where a1, ..., ar are separable over k.

• (iii) In (ii), one can take r = 1.

• (iv) If K is an algebraic closure of k, then there are precisely [F : k] embeddings F into K which fix k.

• (v) If K is any normal extension of k such that F embeds into K in at least one way, then there are precisely[F : k] embeddings F into K which fix k.

In the above, (iii) is known as the primitive element theorem.Fix the algebraic closure k , and denote by ks the set of all elements of k that are separable over k. ks is then separablealgebraic over k and any separable algebraic subextension of k is contained in ks ; it is called the separable closureof k (inside k ). k is then purely inseparable over ks . Put in another way, k is perfect if and only if k = ks .

14.6 Differential criteria

The separability can be studied with the aid of derivations and Kähler differentials. Let F be a finitely generated fieldextension of a field k . Then

dimF Derk(F, F ) ≥ tr. degk F

where the equality holds if and only if F is separable over k.In particular, if F/k is an algebraic extension, then Derk(F, F ) = 0 if and only if F/k is separable.[26]

Let D1, ..., Dm be a basis of Derk(F, F ) and a1, ..., am ∈ F . Then F is separable algebraic over k(a1, ..., am) ifand only if the matrix Di(aj) is invertible. In particular, when m = tr. degk F , {a1, ..., am} above is called theseparating transcendence basis.

14.7 See also• Purely inseparable extension

• Perfect field

• Primitive element theorem

• Normal extension

• Galois extension

• Algebraic closure

14.8 Notes[1] Isaacs, p. 281

[2] Isaacs, Theorem 18.13, p. 282

[3] Isaacs, Theorem 18.11, p. 281

[4] Isaacs, p. 293

[5] Isaacs, p. 298

58 CHAPTER 14. SEPARABLE EXTENSION

[6] Isaacs, p. 280

[7] Isaacs, Lemma 18.10, p. 281

[8] Isaacs, Lemma 18.7, p. 280

[9] Isaacs, Theorem 19.4, p. 295

[10] Isaacs, Corollary 19.5, p. 296

[11] Isaacs, Corollary 19.6, p. 296

[12] Isaacs, Corollary 19.9, p. 298

[13] Isaacs, Theorem 19.7, p. 297

[14] Isaacs, p. 299

[15] Isaacs, Lemma 19.15, p. 300

[16] Isaacs, Corollary 19.17, p. 301

[17] Isaacs, Corollary 18.12, p. 281

[18] Isaacs, Theorem 19.14, p. 300

[19] Isaacs, p. 302

[20] Lang 2002, Corollary V.6.2

[21] Isaacs, Theorem 19.19, p. 302

[22] Isaacs, Lemma 19.20, p. 302

[23] Isaacs, Corollary 19.21, p. 303

[24] The characteristic exponent of k is 1 if k has characteristic zero; otherwise, it is the characteristic of k.

[25] Fried & Jarden (2008) p.38

[26] Fried & Jarden (2008) p.49

14.9 References• Borel, A. Linear algebraic groups, 2nd ed.

• P.M. Cohn (2003). Basic algebra

• Fried,Michael D.; Jarden,Moshe (2008). Field arithmetic. Ergebnisse derMathematik und ihrer Grenzgebiete.3. Folge 11 (3rd ed.). Springer-Verlag. ISBN 978-3-540-77269-9. Zbl 1145.12001.

• I. Martin Isaacs (1993). Algebra, a graduate course (1st ed.). Brooks/Cole Publishing Company. ISBN 0-534-19002-2.

• Kaplansky, Irving (1972). Fields and rings. Chicago lectures in mathematics (Second ed.). University ofChicago Press. pp. 55–59. ISBN 0-226-42451-0. Zbl 1001.16500.

• M. Nagata (1985). Commutative field theory: new edition, Shokado. (Japanese)

• Silverman, Joseph (1993). The Arithmetic of Elliptic Curves. Springer. ISBN 0-387-96203-4.

14.10 External links• Hazewinkel, Michiel, ed. (2001), “separable extension of a field k”, Encyclopedia of Mathematics, Springer,ISBN 978-1-55608-010-4

Chapter 15

Simple extension

In field theory, a simple extension is a field extension which is generated by the adjunction of a single element. Simpleextensions are well understood and can be completely classified.The primitive element theorem provides a characterization of the finite simple extensions.

15.1 Definition

A field extension L/K is called a simple extension if there exists an element θ in L with

L = K(θ).

The element θ is called a primitive element, or generating element, for the extension; we also say that L is generatedover K by θ.Every finite field is a simple extension of the prime field of the same characteristic. More precisely, if p is a primenumber and q = pd the field Fq of q elements is a simple extension of degree d of Fp. This means that it is generatedby an element θ which is a root of an irreducible polynomial of degree d. However, in this case, θ is normally notreferred to as a primitive element.In fact, a primitive element of a finite field is usually defined as a generator of the field’s multiplicative group. Moreprecisely, by little Fermat theorem, the nonzero elements of Fq (i.e. its multiplicative group) are the roots of theequation

xq−1 − 1 = 0,

that is the (q−1)-th roots of unity. Therefore, in this context, a primitive element is a primitive (q−1)-th root ofunity, that is a generator of the multiplicative group of the nonzero elements of the field. Clearly, a group primitiveelement is a field primitive element, but the contrary is false.Thus the general definition requires that every element of the field may be expressed as a polynomial in the generator,while, in the realm of finite fields, every nonzero element of the field is a pure power of the primitive element. Todistinguish these meanings one may use field primitive element of L over K for the general notion, and groupprimitive element for the finite field notion.[1]

15.2 Structure of simple extensions

If L is a simple extension of K generated by θ, it is the only field contained in L which contains both K and θ.This means that every element of L can be obtained from the elements of K and θ by finitely many field operations(addition, subtraction, multiplication and division).

59

60 CHAPTER 15. SIMPLE EXTENSION

Let us consider the polynomial ring K[X]. One of its main properties is that there exists a unique ring homomorphism

φ : K[X] → L

p(X) 7→ p(θ) .

Two cases may occur.If φ is injective, it may be extended to the field of fractions K(X) of K[X]. As we have supposed that L is generatedby θ, this implies that φ is an isomorphism from K(X) onto L. This implies that every element of L is equal to anirreducible fraction of polynomials in θ, and that two such irreducible fractions are equal if and only if one may passfrom one to the other by multiplying the numerator and the denominator by the same non zero element of K.If φ is not injective, let p(X) be a generator of its kernel, which is thus the minimal polynomial of θ. The imageof φ is a subring of L, and thus an integral domain. This implies that p is an irreducible polynomial, and thus thatthe quotient ring K[X]/⟨p⟩ is a field. As L is generated by θ, φ is surjective, and φ induces an isomorphism fromK[X]/⟨p⟩ onto L. This implies that every element of L is equal to a unique polynomial in θ, of degree lower thanthe degree of the extension.

15.3 Examples• C:R (generated by i)

• Q(√2):Q (generated by √2), more generally any number field (i.e., a finite extension ofQ) is a simple extensionQ(α) for some α. For example, Q(

√3,√7) is generated by

√3 +

√7 .

• F(X):F (generated by X).

15.4 References[1] (Roman 1995)

• Roman, Steven (1995). Field Theory. Graduate Texts inMathematics 158. NewYork: Springer-Verlag. ISBN0-387-94408-7. Zbl 0816.12001.

Chapter 16

Splitting field

In abstract algebra, a splitting field of a polynomial with coefficients in a field is a smallest field extension of thatfield over which the polynomial splits or decomposes into linear factors.

16.1 Definition

A splitting field of a polynomial p(X) over a field K is a field extension L of K over which p factors into linear factors

p(X) =∏deg(p)

i=1 (X − ai) where for each i we have (X − ai) ∈ L[X]

and such that the roots ai generate L over K. The extension L is then an extension of minimal degree over K in whichp splits. It can be shown that such splitting fields exist and are unique up to isomorphism. The amount of freedom inthat isomorphism is known as the Galois group of p (if we assume it is separable).

16.2 Facts

An extension L which is a splitting field for a set of polynomials p(X) over K is called a normal extension of K.Given an algebraically closed field A containing K, there is a unique splitting field L of p between K and A, generatedby the roots of p. If K is a subfield of the complex numbers, the existence is immediate. On the other hand, theexistence of algebraic closures in general is often proved by 'passing to the limit' from the splitting field result, whichtherefore requires an independent proof to avoid circular reasoning.Given a separable extension K′ of K, a Galois closure L of K′ is a type of splitting field, and also a Galois extensionof K containing K′ that is minimal, in an obvious sense. Such a Galois closure should contain a splitting field for allthe polynomials p over K that are minimal polynomials over K of elements a of K′.

16.3 Constructing splitting fields

16.3.1 Motivation

Finding roots of polynomials has been an important problem since the time of the ancient Greeks. Some polynomials,however, such as X2+1 over R, the real numbers, have no roots . By constructing the splitting field for such apolynomial one can find the roots of the polynomial in the new field.

16.3.2 The construction

LetF be a field and p(X) be a polynomial in the polynomial ring F[X] of degree n. The general process for constructingK, the splitting field of p(X) over F, is to construct a sequence of fields F = K0,K1, . . .Kr−1,Kr = K such that

61

62 CHAPTER 16. SPLITTING FIELD

Ki is an extension of Ki₋₁ containing a new root of p(X). Since p(X) has at most n roots the construction will requireat most n extensions. The steps for constructing Ki are given as follows:

• Factorize p(X) over Ki into irreducible factors f1(X)f2(X) · · · fk(X) .

• Choose any nonlinear irreducible factor f(X) = fi(X).

• Construct the field extension Ki₊₁ of Ki as the quotient ring Ki₊₁ = Ki[X]/(f(X)) where (f(X)) denotes the idealin Ki[X] generated by f(X)

• Repeat the process for Ki₊₁ until p(X) completely factors.

The irreducible factor fi used in the quotient construction may be chosen arbitrarily. Although different choices offactors may lead to different subfield sequences the resulting splitting fields will be isomorphic.Since f(X) is irreducible, (f(X)) is a maximal ideal and hence Ki[X]/(f(X)) is, in fact, a field. Moreover, if we letπ : Ki[X] → Ki[X]/(f(X)) be the natural projection of the ring onto its quotient then

f(π(X)) = π(f(X)) = f(X) mod f(X) = 0

so π(X) is a root of f(X) and of p(X).The degree of a single extension [Ki+1 : Ki] is equal to the degree of the irreducible factor f(X). The degree of theextension [K : F] is given by [Kr : Kr−1] · · · [K2 : K1][K1 : F ] and is at most n!.

16.3.3 The field Ki[X]/(f(X))

As mentioned above, the quotient ring Ki₊₁ = Ki[X]/(f(X)) is a field when f(X) is irreducible. Its elements are of theform

cn−1αn−1 + cn−2α

n−2 + · · ·+ c1α+ c0

where the cj are in Ki and α = π(X). (If one considers Ki₊₁ as a vector space over Ki then the powers αj for 0 ≤ j ≤n−1 form a basis.)The elements of Ki₊₁ can be considered as polynomials in α of degree less than n. Addition in Ki₊₁ is given by therules for polynomial addition and multiplication is given by polynomial multiplication modulo f(X). That is, for g(α)and h(α) in Ki₊₁ the product g(α)h(α) = r(α) where r(X) is the remainder of g(X)h(X) divided by f(X) in Ki[X].The remainder r(X) can be computed through long division of polynomials, however there is also a straightforwardreduction rule that can be used to compute r(α) = g(α)h(α) directly. First let

f(X) = Xn + bn−1Xn−1 + · · ·+ b1X + b0.

The polynomial is over a field so one can take f(X) to be monic without loss of generality. Now α is a root of f(X),so

αn = −(bn−1αn−1 + · · ·+ b1α+ b0).

If the product g(α)h(α) has a term αm with m ≥ n it can be reduced as follows:

αnαm−n = −(bn−1α

n−1 + · · ·+ b1α+ b0)αm−n = −

(bn−1α

m−1 + · · ·+ b1αm−n+1 + b0α

m−n)

As an example of the reduction rule, take Ki =Q[X], the ring of polynomials with rational coefficients, and take f(X)= X7 − 2. Let g(α) = α5 + α2 and h(α) = α3 +1 be two elements of Q[X]/(X7 − 2). The reduction rule given byf(X) is α7 = 2 so

g(α)h(α) =(α5 + α2

) (α3 + 1

)= α8 + 2α5 + α2 =

(α7)α+ 2α5 + α2 = 2α5 + α2 + 2α.

16.4. EXAMPLES 63

16.4 Examples

16.4.1 The complex numbers

Consider the polynomial ring R[x], and the irreducible polynomial x2 + 1. The quotient ring R[x] / (x2 + 1) is givenby the congruence x2 ≡ −1. As a result, the elements (or equivalence classes) of R[x] / (x2 + 1) are of the form a +bx where a and b belong to R. To see this, note that since x2 ≡ −1 it follows that x3 ≡ −x, x4 ≡ 1, x5 ≡ x, etc.; and so,for example p + qx + rx2 + sx3 ≡ p + qx + r⋅(−1) + s⋅(−x) = (p − r) + (q − s)⋅x.The addition and multiplication operations are given by firstly using ordinary polynomial addition and multiplication,but then reducing modulo x2 + 1, i.e. using the fact that x2 ≡ −1, x3 ≡ −x, x4 ≡ 1, x5 ≡ x, etc. Thus:

(a1 + b1x) + (a2 + b2x) = (a1 + a2) + (b1 + b2)x,

(a1 + b1x)(a2 + b2x) = a1a2 + (a1b2 + b1a2)x+ (b1b2)x2 ≡ (a1a2 − b1b2) + (a1b2 + b1a2)x .

If we identify a + bx with (a,b) then we see that addition and multiplication are given by

(a1, b1) + (a2, b2) = (a1 + a2, b1 + b2),

(a1, b1) · (a2, b2) = (a1a2 − b1b2, a1b2 + b1a2).

We claim that, as a field, the quotient R[x] / (x2 + 1) is isomorphic to the complex numbers, C. A general complexnumber is of the form a + ib, where a and b are real numbers and i2 = −1. Addition and multiplication are given by

(a1 + ib1) + (a2 + ib2) = (a1 + a2) + i(b1 + b2),

(a1 + ib1) · (a2 + ib2) = (a1a2 − b1b2) + i(a1b2 + a2b1).

If we identify a + ib with (a,b) then we see that addition and multiplication are given by

(a1, b1) + (a2, b2) = (a1 + a2, b1 + b2),

(a1, b1) · (a2, b2) = (a1a2 − b1b2, a1b2 + b1a2) .

The previous calculations show that addition and multiplication behave the same way in R[x] / (x2 + 1) and C. Infact, we see that the map between R[x]/(x2 + 1) and C given by a + bx→ a + ib is a homomorphism with respect toaddition and multiplication. It is also obvious that the map a + bx→ a + ib is both injective and surjective; meaningthat a + bx → a + ib is a bijective homomorphism, i.e. an isomorphism. It follows that, as claimed: R[x] / (x2 + 1)≅ C.

16.4.2 Cubic example

Let K be the rational number field Q and

p(X) = X3 − 2.

Each root of p equals 3√2 times a cube root of unity. Therefore, if we denote the cube roots of unity by

ω1 = 1

ω2 = −1

2+

√3

2i,

ω3 = −1

2−

√3

2i.

64 CHAPTER 16. SPLITTING FIELD

any field containing two distinct roots of p will contain the quotient between two distinct cube roots of unity. Such aquotient is a primitive cube root of unity—either ω2 or ω3 = 1/ω2 . It follows that a splitting field L of p will containω2, as well as the real cube root of 2; conversely, any extension of Q containing these elements contains all the rootsof p. Thus

L = Q(3√2, ω2) = {a+ bω2 + c

3√2 + d

3√2ω2 + e

3√22 + f

3√22ω2 | a, b, c, d, e, f ∈ Q}

Note that applying the construction process outlined in the previous section to this example, one begins withK0 = Qand constructs the field K1 = Q[X]/(X3 − 2) . This field is not the splitting field, but contains one (any) root.However, the polynomial Y 3 − 2 is not irreducible overK1 and in fact, factorizes into (Y −X)(Y 2 +XY +X2) .Note thatX is not an indeterminate, and is in fact an element ofK1 . Now, continuing the process, we obtainK2 =K1[Y ]/(Y 2+XY +X2)which is indeed the splitting field (and is spanned by theQ -basis {1, X,X2, Y,XY,X2Y }).

16.4.3 Other examples

• The splitting field of x2 + 1 over F7 is F49; the polynomial has no roots in F7, i.e., −1 is not a square there,because 7 is not equivalent to 1 (mod 4).[1]

• The splitting field of x2 − 1 over F7 is F7 since x2 − 1 = (x + 1)(x − 1) already factors into linear factors.

• We calculate the splitting field of f(x) = x3 + x + 1 over F2. It is easy to verify that f(x) has no roots in F2,hence f(x) is irreducible in F2[x]. Put r = x + (f(x)) in F2[x]/(f(x)) so F2(r) is a field and x3 + x + 1 = (x+ r)(x2 + ax + b) in F2(r)[x]. Note that we can write + for − since the characteristic is two. Comparison ofcoefficients shows that a = r and b = 1 + r2. The elements of F2(r) can be listed as c + dr + er2, where c, d, eare in F2. There are eight elements: 0, 1, r, 1 + r, r2, 1 + r2, r + r2 and 1 + r + r2. Substituting these in x2 +rx + 1 + r2 we reach (r2)2 + r(r2) + 1 + r2 = r4 + r3 + 1 + r2 = 0, therefore x3 + x + 1 = (x + r)(x + r2)(x + (r+ r2)) for r in F2[x]/(f(x)); E = F2(r) is a splitting field of x3 + x + 1 over F2.

16.5 See also• Rupture field

16.6 Notes[1] Instead of applying this characterization of odd prime moduli for which −1 is a square, one could just check that the set of

squares in F7 is the set of classes of 0, 1, 4, and 2, which does not include the class of −1≡6.

16.7 References• Dummit, David S., and Foote, Richard M. (1999). Abstract Algebra (2nd ed.). New York: JohnWiley & Sons,Inc. ISBN 0-471-36857-1.

• Hazewinkel, Michiel, ed. (2001), “Splitting field of a polynomial”, Encyclopedia of Mathematics, Springer,ISBN 978-1-55608-010-4

• Weisstein, Eric W., “Splitting field”, MathWorld.

Chapter 17

Tower of fields

In mathematics, a tower of fields is a sequence of field extensions

F0 ⊆ F1 ⊆ ... ⊆ Fn ⊆ ...

The name comes from such sequences often being written in the form

...|F2

|F1

|F0.

A tower of fields may be finite or infinite.

17.1 Examples• Q ⊆ R ⊆ C is a finite tower with rational, real and complex numbers.

• The sequence obtained by letting F0 be the rational numbers Q, and letting

Fn+1 = Fn

(21/2

n)

(i.e. Fn₊₁ is obtained from Fn by adjoining a 2n th root of 2) is an infinite tower.

• If p is a prime number the p th cyclotomic tower ofQ is obtained by letting F0 =Q and Fn be the field obtainedby adjoining to Q the pn th roots of unity. This tower is of fundamental importance in Iwasawa theory.

• The Golod–Shafarevich theorem shows that there are infinite towers obtained by iterating the Hilbert class fieldconstruction to a number field.

17.2 References• Section 4.1.4 of Escofier, Jean-Pierre (2001), Galois theory, Graduate Texts in Mathematics 204, Springer-Verlag, ISBN 978-0-387-98765-1

65

66 CHAPTER 17. TOWER OF FIELDS

17.3 Text and image sources, contributors, and licenses

17.3.1 Text• Abelian extension Source: https://en.wikipedia.org/wiki/Abelian_extension?oldid=702527109 Contributors: Zundark, Michael Hardy,

Looxix~enwiki, Schneelocke, Revolver, Charles Matthews, Jni, GeneWard Smith, Vivacissamamente, Oleg Alexandrov, R.e.b., Mathbot,SmackBot, CRGreathouse, RobHar, Dugwiki, Eleuther, Vanish2, David Eppstein, Ideal gas equation, Addbot, Luckas-bot, ArthurBot,AmphBot, ChuispastonBot, Enyokoyama and Anonymous: 7

• Algebraic closure Source: https://en.wikipedia.org/wiki/Algebraic_closure?oldid=696129814Contributors: DamianYerrick, AxelBoldt,Mav, Zundark, Enchanter, Michael Hardy, Charles Matthews, Dysprosia, Tobias Bergemann, Giftlite, DefLog~enwiki, Vivacissama-mente, Rich Farmbrough, Haham hanuka, HasharBot~enwiki, Oleg Alexandrov, FlaBot, YurikBot, Gslin, SmackBot, Jushi, Bluebot,Gutworth, NeilFraser, MvH, RekishiEJ, Gregbard, Salgueiro~enwiki, JAnDbot, David Eppstein, Aram33~enwiki, DorganBot, Kriega,Ideal gas equation, MystBot, Legobot, Luckas-bot, Yobot, AnomieBOT, Howard McCay, Stickelberger, MaximalIdeal, Spaetzle, Kodip,Deltahedron, SantiLak and Anonymous: 23

• Algebraic extension Source: https://en.wikipedia.org/wiki/Algebraic_extension?oldid=653316776 Contributors: AxelBoldt, Zundark,SimonP, Alodyne, TakuyaMurata, Charles Matthews, Dysprosia, MathMartin, Giftlite, Sim~enwiki, Wmahan, DefLog~enwiki, Vina,Klemen Kocjancic, Rich Farmbrough, Andi5, EmilJ, Mdd, Drbreznjev, Oleg Alexandrov, Banus, SmackBot, Maksim-e~enwiki, Uny-oyega, MalafayaBot, Cícero, Vina-iwbot~enwiki, Will Beback, MvH, Jim.belk, Mets501, CmdrObot, CBM, Thijs!bot, RobHar, Vanish2,Trumpet marietta 45750, Linefeed, JackSchmidt, Ideal gas equation, He7d3r, Hans Adler, Legobot, Luckas-bot, Sz-iwbot, Telementor,RjwilmsiBot, EmausBot, Wcherowi, ChrisGualtieri, Deltahedron, Bg9989, K9re11 and Anonymous: 14

• Characteristic (algebra) Source: https://en.wikipedia.org/wiki/Characteristic_(algebra)?oldid=680705935Contributors: AxelBoldt, LC~enwiki,Bryan Derksen, Zundark, Michael Hardy, Revolver, Charles Matthews, Steinsky, JensMueller, Moink, Jleedev, Tosha, Giftlite, Ben-FrantzDale, Fropuff, Dratman, Rpchase, Barnaby dawson, Guanabot, Pjacobi, MuDavid, Paul August, Rgdboer, Emvee~enwiki, Joriki,LOL, Graham87, Salix alba, YurikBot, Pred, GrinBot~enwiki, Eskimbot, Mhss, Nbarth, Lesnail, Vina-iwbot~enwiki, Xtv, Satori Son,W3asal, RobHar, JAnDbot, David Eppstein, JoergenB, STBot, Sigmundur, VolkovBot, LokiClock, TXiKiBoT, Kyle Pena, Finlux, SieBot,YonaBot, SimonTrew, Malatinszky, Anchor Link Bot, ArdClose, Niceguyedc, BOTarate, Addbot, Zorrobot, Luckas-bot, AnomieBOT,Rckrone, Javasava, ZéroBot, Offsure, Quondum, BG19bot, RuHouse'ls, ChrisGualtieri, GeoffreyT2000 and Anonymous: 25

• Degree of a field extension Source: https://en.wikipedia.org/wiki/Degree_of_a_field_extension?oldid=699773138 Contributors: Axel-Boldt, MathMartin, Goochelaar, Arthena, Oleg Alexandrov, Dmharvey, Mathaxiom~enwiki, Bluebot, Jim.belk, RobHar, Yoda of Borg,Ideal gas equation, SuperHamster, Addbot, Glane23, Citation bot, Devingragg, EmausBot, WikitanvirBot, 28bot, Braincricket, Markviking and Anonymous: 6

• Dual basis in a field extension Source: https://en.wikipedia.org/wiki/Dual_basis_in_a_field_extension?oldid=644686663 Contributors:Charles Matthews, Gene Ward Smith, CryptoDerk, ArnoldReinhold, SmackBot, Ideal gas equation, Niceguyedc, Erik9bot, Brad7777,LimeyCinema1960 and Anonymous: 1

• Field extension Source: https://en.wikipedia.org/wiki/Field_extension?oldid=685545830 Contributors: AxelBoldt, Zundark, Edward,TakuyaMurata, Daran, Naddy, Lowellian, MathMartin, Giftlite, Fropuff, Mazi, El C, Bookofjude, EmilJ, Oleg Alexandrov, Marudub-shinki, Graham87, FlaBot, YurikBot, Dmharvey, Mathaxiom~enwiki, Grubber, KnightRider~enwiki, SmackBot, Nbarth, Foxjwill, Ewjw,Jim.belk, Madmath789, CRGreathouse, Thijs!bot, RobHar, Escarbot, JAnDbot, Magioladitis, David Eppstein, Cpiral, Policron, STBotD,PerezTerron, TXiKiBoT, Don4of4, AlleborgoBot, Cwkmail, JackSchmidt, Yasmar, Ideal gas equation, Mpd1989, Alexbot, He7d3r, Ad-dbot, PV=nRT, Ptbotgourou, Calle, Xqbot, Point-set topologist, Uuo, Sławomir Biały, Vanished user fijtji34toksdcknqrjn54yoimascj,Cenkner, YFdyh-bot and Anonymous: 47

• Finite field Source: https://en.wikipedia.org/wiki/Finite_field?oldid=701946079Contributors: AxelBoldt, Bryan Derksen, Zundark, Tar-quin, Toby Bartels, PierreAbbat, Chas zzz brown, Michael Hardy, TakuyaMurata, Karada, J-Wiki, Revolver, Charles Matthews, Dcoetzee,Joshuabowman, Dysprosia, Fibonacci, Fredrik, Ojigiri~enwiki, Bkell, Wikibot, Tobias Bergemann, Giftlite, Gene Ward Smith, Lethe,Fropuff, Dratman, Waltpohl, Dries~enwiki, Pmanderson, Elroch, Karl Dickman, Andreas Kaufmann, Vivacissamamente, Mike Rosoft,Paul August, Petrus~enwiki, Zaslav, Army1987, Giraffedata, Atlant, Emvee~enwiki, Oleg Alexandrov, RHaworth, Ma Baker, Hyper-cube~enwiki, Isnow, HannsEwald, OmriSegal, Chobot, Bgwhite, Algebraist, YurikBot, JWB, Dmharvey, Lenthe, KSmrq, DYLANLENNON~enwiki, Hv, SmackBot, Reedy, Zeycus, Gilliam, Chris the speller, Bluebot, Alan smithee, Nbarth, Daqu, BlackFingolfin,Dgessner, MvH, JoshuaZ, Noegenesis, Schildt.a, WAREL, VoiceOfOdin, Az1568, Floridi~enwiki, Shernren, Thijs!bot, A3RO, Rob-Har, Klausness, Urdutext, AntiVandalBot, Lovibond, Magioladitis, Vanish2, JamesBWatson, Albmont, Brusegadi, David Eppstein, Jo-ergenB, Cpiral, Maproom, Jacksonwalters, VolkovBot, Safemariner, TXiKiBoT, Oshwah, MichaelShoemaker, SieBot, Thehotelambush,JackSchmidt, Yoda of Borg, Mild Bill Hiccup, Auntof6, 800km3rk, Bender2k14, Sun Creator, SchreiberBike, MagnusPI, Leonid 01, Sah-mosavian1, SilvonenBot, MystBot, Addbot, Download, LinkFA-Bot, Legobot, Luckas-bot, Yobot, AnomieBOT, Greggles612, ArthurBot,LilHelpa, Drilnoth, Etoombs, Anne Bauval, Point-set topologist, Sonoluminesence, FrescoBot, Turbanoff, RedAcer, AaronEmi, Doublesharp, Bj norge, Racerx11, Lfrazier11, Quondum, D.Lazard, Wcherowi, Udcep, Hannob, Frietjes, Joel B. Lewis, BG19bot, Ikamusume-Fan, DNarvaez, MathKnight-at-TAU, Dexbot, Citizentoad, Jamesmath, Cyrapas, K9re11, Spencer m67, Velociraptor 11235813, Rkinser,Chumpih, Teddyktchan, Dyott, GeoffreyT2000, Parkerf, Joseph2302, Some1Redirects4You and Anonymous: 119

• Galois extension Source: https://en.wikipedia.org/wiki/Galois_extension?oldid=674571545 Contributors: Edward, Charles Matthews,MathMartin, Fuelbottle, Giftlite, EmilJ, Mdd, Algebraist, Dmharvey, Greatal386, SmackBot, Eskimbot, Gutworth, RyanEberhart, MvH,Thijs!bot, RobHar, Etale, TomyDuby, Sigmundur, TXiKiBoT, Omerks, Dogah, Cwkmail, Ideal gas equation, Alexbot, Bender2k14,Addbot, Luckas-bot, Xqbot, Point-set topologist, RibotBOT, Erik9bot, Anita5192, Solomon7968, Enyokoyama, Brirush and Anonymous:9

• Irreducible polynomial Source: https://en.wikipedia.org/wiki/Irreducible_polynomial?oldid=693736482 Contributors: AxelBoldt, Tar-quin, TakuyaMurata, Angela, Yaakov~enwiki, Vargenau, Loren Rosen, Charles Matthews, Dcoetzee, Molinari, Dysprosia, Zoicon5,Fredrik, Gandalf61, Giftlite, Dratman, Macrakis, CryptoDerk, TheObtuseAngleOfDoom, Sam Derbyshire, El C, 3mta3, Culix, Arneth,Madmardigan53, Rjwilmsi, Marozols, Michael Slone, Marc Harper, SmackBot, Mmernex, Adam majewski, Eskimbot, Mdd4696, Alansmithee, Sct72, Wewe (de Cádiz), Allansteel, Mets501, Pagh, Madmath789, CRGreathouse, Cisco Systems, Kilva, Dogaroon, MashiahDavidson, .anacondabot, Jakob.scholbach, David Eppstein, Boston, ReturnKeyandShiftKey, TXiKiBoT, Uwhoff, Wpoely86, Mild BillHiccup, Bender2k14, Marc van Leeuwen, Addbot, James.robinson, Zorrobot, Yobot, AnomieBOT, Point-set topologist, TLange, Foo-barnix, Duoduoduo, EmausBot, ZéroBot, D.Lazard, Odysseus1479, ClueBot NG, Erick GR, Helpful Pixie Bot, Dexbot, NoKo, Kahtar,Monkbot, Loraof and Anonymous: 48

17.3. TEXT AND IMAGE SOURCES, CONTRIBUTORS, AND LICENSES 67

• Normal extension Source: https://en.wikipedia.org/wiki/Normal_extension?oldid=689628071Contributors: AxelBoldt, Zundark,MichaelHardy, MathKnight, WalterM, Art LaPella, EmilJ, Oleg Alexandrov, Bgwhite, Michael Slone, Russell C. Sibley, Maksim-e~enwiki, JC-Santos, Gutworth, Khazar, Timmie.merc, Thijs!bot, Konradek, RobHar, Jakob.scholbach, LokiClock, Kyle the bot, Plclark, COBot, Idealgas equation, Alexbot, Bender2k14, Sandrobt, Addbot, Expz, Luckas-bot, Yobot, Jim1138, FrescoBot, TobeBot, Josve05a, Helpful PixieBot and Anonymous: 12

• Quotient ring Source: https://en.wikipedia.org/wiki/Quotient_ring?oldid=666100333 Contributors: AxelBoldt, Patrick, Michael Hardy,Ciphergoth, MathMartin, Nikitadanilov, Giftlite, Waltpohl, Jorge Stolfi, Rgdboer, Linas, Marudubshinki, Chobot, Algebraist, YurikBot,Rsrikanth05, That Guy, From That Show!, Reedy, MalafayaBot, Jim.belk, Rschwieb, CRGreathouse, CBM, Thijs!bot, Albmont, Policron,DemonicInfluence, VolkovBot, Arcfrk, Katzmik, SieBot, Addbot, Jarble, Luckas-bot, Ark11, AnomieBOT, Measles, Ebony Jackson,RjwilmsiBot, EmausBot, ZéroBot, Otaria, Noahc66260, ChrisGualtieri, Mathedu and Anonymous: 19

• Root of unity Source: https://en.wikipedia.org/wiki/Root_of_unity?oldid=702657899 Contributors: AxelBoldt, The Anome, Tarquin,Patrick, Michael Hardy, GTBacchus, Oyd11, Stevenj, Schneelocke, Charles Matthews, Dysprosia, Hyacinth, Yoheythere, Wilke, Gan-dalf61, MathMartin, Giftlite, Dratman, Yath, Fangz, Doops, Elroch, Vivacissamamente, Zaslav, EmilJ, Nandhp, Msh210, Velella,H2g2bob, Kenyon, Mcsee, Alexrudd, Linas, Mindmatrix, Shreevatsa, Isnow, Marudubshinki, HannsEwald, Nneonneo, Bubba73, FlaBot,VKokielov, RexNL, Glenn L, YurikBot, Gaius Cornelius, Arichnad, Netrapt, Fractalchez, Bo Jacoby, Edin1, SmackBot, Incnis Mrsi,InverseHypercube, Melchoir, Eskimbot, Betacommand, Jushi, Bird of paradox, Silly rabbit, Nbarth, DRLB, Tesseran, J. Finkelstein,JoshuaZ, Loadmaster, Madmath789, Eastlaw, CRGreathouse, HenningThielemann, Myasuda, Cydebot, Zahlentheorie, RobHar, Sher-brooke, CZeke, Turgidson, Dcooper, Kerotan, Magioladitis, David Eppstein, JoergenB, QuantumGroupie, DavidCBryant, VolkovBot,Saziel, Hesam7, Dmcq, Omerks, YonaBot, Tommyjs, SamGonshaw, Anchor Link Bot, Mild Bill Hiccup, Alexbot, Bender2k14, Dule-orlovic, Virginia-American, CàlculIntegral, Tangi-tamma, Addbot, Fgnievinski, Download, LaaknorBot, Gail, Luckas-bot, Yobot, AnomieBOT,Citation bot, Devoutb3nji, Anne Bauval, FancyMouse, Howard McCay, FrescoBot, Citation bot 1, WikitanvirBot, ZéroBot, D.Lazard,Quandle, Chrgue, Mesoderm, Helpful Pixie Bot, Climb026, Deltahedron, Jamesx12345, Csjacobs24, Amortias, Loraof and Anonymous:78

• Separable extension Source: https://en.wikipedia.org/wiki/Separable_extension?oldid=700598244 Contributors: AxelBoldt, Zundark,Edward, TakuyaMurata, Charles Matthews, Dysprosia, Vivacissamamente, Shotwell, Mazi, Bender235, Vipul, EmilJ, Keenan Pepper,OlegAlexandrov, R.e.b., YurikBot, SmackBot, Eskimbot, Mets501,WLior, Tac-Tics, Dragonflare82, Thijs!bot, RobHar, Jakob.scholbach,Etale, Allispaul, LokiClock,Wedhorn, Ideal gas equation, Niceguyedc, Alexbot, Bender2k14, Addbot, Luckas-bot, FredrikMeyer, AnomieBOT,Citation bot, LilHelpa, GrouchoBot, Point-set topologist, Hkhk59333, Citation bot 1, John of Reading, Codygunton, D.Lazard, Ben-delacBOT, Deltahedron, Mark viking, Gjbayes and Anonymous: 23

• Simple extension Source: https://en.wikipedia.org/wiki/Simple_extension?oldid=595508851 Contributors: TakuyaMurata, MathMartin,Giftlite, Marudubshinki, Salix alba, YurikBot, Dmharvey, Michael Slone, Eskimbot, PetrMatas, Vanish2, Hesam7, JackSchmidt, Ideal gasequation, NuclearWarfare, Sandrobt, MystBot, Addbot, TobeBot, D.Lazard, JordiGH, Wcherowi, Deltahedron, Darvii and Anonymous:5

• Splitting field Source: https://en.wikipedia.org/wiki/Splitting_field?oldid=703608186 Contributors: Michael Hardy, Looxix~enwiki,Charles Matthews, Giftlite, Rajsekar, EmilJ, LutzL, Burn, Oleg Alexandrov, Mandarax, Dmharvey, Incnis Mrsi, Eskimbot, Jprg1966,MvH, TooMuchMath, Harej bot, JAnDbot, GromXXVII, .anacondabot, Magioladitis, TomyDuby, Policron, LokiClock, Jobu0101, Co-dairem, SurJector, He7d3r, Bender2k14, Marc van Leeuwen, Addbot, Dyaa, PV=nRT, Yobot, TaBOT-zerem, AnomieBOT, Cpryby,Jujutacular, Greenfernglade, Pgdoyle, EmausBot, Fly by Night, Razghandi, Nosuchforever, Lundril, Nigellwh, Mdavis94538 and Anony-mous: 27

• Tower of fields Source: https://en.wikipedia.org/wiki/Tower_of_fields?oldid=632934103 Contributors: Pol098, Crasshopper, RobHar,Quondum, Technopop.tattoo, K9re11 and Anonymous: 1

17.3.2 Images• File:3rd_roots_of_unity.svg Source: https://upload.wikimedia.org/wikipedia/commons/3/39/3rd_roots_of_unity.svg License: Public

domain Contributors: Based on File:3rd-roots-of-unity.png by User:MarekSchmidt Original artist: Nandhp• File:Complex_x_hoch_3.jpg Source: https://upload.wikimedia.org/wikipedia/commons/2/2a/Complex_x_hoch_3.jpg License: Public

domain Contributors: made with mathematica, own work Original artist: Jan Homann• File:Complex_x_hoch_5.jpg Source: https://upload.wikimedia.org/wikipedia/commons/0/08/Complex_x_hoch_5.jpg License: Public

domain Contributors: made with mathematica, own work Original artist: Jan Homann• File:Edit-clear.svg Source: https://upload.wikimedia.org/wikipedia/en/f/f2/Edit-clear.svg License: Public domain Contributors: TheTango! Desktop Project. Original artist:The people from the Tango! project. And according to themeta-data in the file, specifically: “Andreas Nilsson, and Jakub Steiner (althoughminimally).”

• File:One5Root.svg Source: https://upload.wikimedia.org/wikipedia/commons/4/40/One5Root.svg License: CC BY-SA 3.0 Contribu-tors: Own work Original artist: Loadmaster (David R. Tribble)

• File:Question_book-new.svg Source: https://upload.wikimedia.org/wikipedia/en/9/99/Question_book-new.svg License: Cc-by-sa-3.0Contributors:Created from scratch in Adobe Illustrator. Based on Image:Question book.png created by User:Equazcion Original artist:Tkgd2007

• File:Roots_of_unity,_golden_ratio.svg Source: https://upload.wikimedia.org/wikipedia/commons/a/af/Roots_of_unity%2C_golden_ratio.svg License: CC0 Contributors: Own work Original artist: Incnis Mrsi

• File:Star_polygon_8-2.svg Source: https://upload.wikimedia.org/wikipedia/commons/7/76/Star_polygon_8-2.svg License: Public do-main Contributors: Own work Original artist: Inductiveload

• File:Text_document_with_red_question_mark.svg Source: https://upload.wikimedia.org/wikipedia/commons/a/a4/Text_document_with_red_question_mark.svg License: Public domain Contributors: Created by bdesham with Inkscape; based upon Text-x-generic.svgfrom the Tango project. Original artist: Benjamin D. Esham (bdesham)

• File:Wikibooks-logo-en-noslogan.svg Source: https://upload.wikimedia.org/wikipedia/commons/d/df/Wikibooks-logo-en-noslogan.svg License: CC BY-SA 3.0 Contributors: Own work Original artist: User:Bastique, User:Ramac et al.

68 CHAPTER 17. TOWER OF FIELDS

17.3.3 Content license• Creative Commons Attribution-Share Alike 3.0