Heyting Field

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Chapter 1

Apartness relation

“Apart” redirects here. For the song by The Cure, see Wish (The Cure album). For the 2011 film, see Apart (film).

In constructive mathematics, an apartness relation is a constructive form of inequality, and is often taken to be morebasic than equality. It is often written as # to distinguish from the negation of equality (the denial inequality) ≠, whichis weaker.

1.1 Description

An apartness relation is a symmetric irreflexive binary relation with the additional condition that if two elements areapart, then any other element is apart from at least one of them (this last property is often called co-transitivity orcomparison).That is, a binary relation # is an apartness relation if it satisfies:[1]

1. ¬ (x#x)

2. x#y → y#x

3. x#y → (x#z ∨ y#z)

The negation of an apartness relation is an equivalence relation, as the above three conditions become reflexivity,symmetry, and transitivity. If this equivalence relation is in fact equality, then the apartness relation is called tight.That is, # is a tight apartness relation if it additionally satisfies:

¬ (x#y) → x = y

In classical mathematics, it also follows that every apartness relation is the negation of an equivalence relation, andthe only tight apartness relation on a given set is the negation of equality. So in that domain, the concept is not useful.In constructive mathematics, however, this is not the case.The prototypical apartness relation is that of the real numbers: two real numbers are said to be apart if there exists(one can construct) a rational number between them. In other words, real numbers x and y are apart if there existsa rational number z such that x < z < y or y < z < x. The natural apartness relation of the real numbers is then thedisjunction of its natural pseudo-order. The complex numbers, real vector spaces, and indeed any metric space thennaturally inherit the apartness relation of the real numbers, even though they do not come equipped with any naturalordering.If there is no rational number between two real numbers, then the two real numbers are equal. Classically, then, iftwo real numbers are not equal, one would conclude that there exists a rational number between them. However itdoes not follow that one can actually construct such a number. Thus to say two real numbers are apart is a strongerstatement, constructively, than to say that they are not equal, and while equality of real numbers is definable in terms

2

1.2. REFERENCES 3

of their apartness, the apartness of real numbers cannot be defined in terms of their equality. For this reason, inconstructive topology especially, the apartness relation over a set is often taken as primitive, and equality is a definedrelation.A set endowed with an apartness relation is known as a constructive setoid. A function f : A → B where A and Bare constructive setoids is called a morphism for #A and #B if ∀x, y : A. f(x) #B f(y) ⇒ x #A y .

1.2 References[1] Troelstra, A. S.; Schwichtenberg, H. (2000), Basic proof theory, Cambridge Tracts in Theoretical Computer Science 43

(2nd ed.), Cambridge University Press, Cambridge, p. 136, doi:10.1017/CBO9781139168717, ISBN 0-521-77911-1, MR1776976.

Chapter 2

Field (mathematics)

This article is about fields in algebra. For fields in geometry, see Vector field. For other uses, see Field (disambigua-tion).

In mathematics, a field is one of the fundamental algebraic structures used in abstract algebra. It is a nonzerocommutative division ring, or equivalently a ring whose nonzero elements form an abelian group under multiplica-tion. As such it is an algebraic structure with notions of addition, subtraction, multiplication, and division satisfyingthe appropriate abelian group equations and distributive law. The most commonly used fields are the field of realnumbers, the field of complex numbers, and the field of rational numbers, but there are also finite fields, algebraicfunction fields, algebraic number fields, p-adic fields, and so forth.Any field may be used as the scalars for a vector space, which is the standard general context for linear algebra. Thetheory of field extensions (including Galois theory) involves the roots of polynomials with coefficients in a field; amongother results, this theory leads to impossibility proofs for the classical problems of angle trisection and squaring thecircle with a compass and straightedge, as well as a proof of the Abel–Ruffini theorem on the algebraic insolubilityof quintic equations. In modern mathematics, the theory of fields (or field theory) plays an essential role in numbertheory and algebraic geometry.As an algebraic structure, every field is a ring, but not every ring is a field. The most important difference is that fieldsallow for division (though not division by zero), while a ring need not possess multiplicative inverses; for example theintegers form a ring, but 2x = 1 has no solution in integers. Also, the multiplication operation in a field is required tobe commutative. A ring in which division is possible but commutativity is not assumed (such as the quaternions) iscalled a division ring or skew field. (Historically, division rings were sometimes referred to as fields, while fields werecalled commutative fields.)As a ring, a field may be classified as a specific type of integral domain, and can be characterized by the following(not exhaustive) chain of class inclusions:

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

2.1 Definition and illustration

Intuitively, a field is a set F that is a commutative group with respect to two compatible operations, addition andmultiplication (the latter excluding zero), with “compatible” being formalized by distributivity, and the caveat that theadditive and the multiplicative identities are distinct (0 ≠ 1).The most common way to formalize this is by defining a field as a set together with two operations, usually calledaddition and multiplication, and denoted by + and ·, respectively, such that the following axioms hold (note thatsubtraction and division are defined in terms of the inverse operations of addition and multiplication):[note 1]

Closure of F under addition and multiplication For all a, b in F, both a + b and a · b are in F (or more formally,+ and · are binary operations on F).

4

2.1. DEFINITION AND ILLUSTRATION 5

Associativity of addition and multiplication For all a, b, and c in F, the following equalities hold: a + (b + c) = (a+ b) + c and a · (b · c) = (a · b) · c.

Commutativity of addition and multiplication For all a and b in F, the following equalities hold: a + b = b + a anda · b = b · a.

Existence of additive and multiplicative identity elements There exists an element ofF, called the additive identityelement and denoted by 0, such that for all a in F, a + 0 = a. Likewise, there is an element, called themultiplicative identity element and denoted by 1, such that for all a in F, a · 1 = a. To exclude the trivial ring,the additive identity and the multiplicative identity are required to be distinct.

Existence of additive inverses and multiplicative inverses For every a in F, there exists an element −a in F, suchthat a + (−a) = 0. Similarly, for any a in F other than 0, there exists an element a−1 in F, such that a · a−1 =1. (The elements a + (−b) and a · b−1 are also denoted a − b and a/b, respectively.) In other words, subtractionand division operations exist.

Distributivity of multiplication over addition For all a, b and c in F, the following equality holds: a · (b + c) = (a ·b) + (a · c).

A field is therefore an algebraic structure F, +, ·, −, −1, 0, 1 ; of type 2, 2, 1, 1, 0, 0 , consisting of two abeliangroups:

• F under +, −, and 0;

• F ∖ {0} under ·, −1, and 1, with 0 ≠ 1,

with · distributing over +.[1]

2.1.1 First example: rational numbers

A simple example of a field is the field of rational numbers, consisting of numbers which can be written as fractionsa/b, where a and b are integers, and b ≠ 0. The additive inverse of such a fraction is simply −a/b, and the multiplicativeinverse (provided that a ≠ 0) is b/a. To see the latter, note that

b

a· ab=

ba

ab= 1.

The abstractly required field axioms reduce to standard properties of rational numbers, such as the law of distributivity

a

b·(c

d+

e

f

)

=a

b·(c

d· ff+

e

f· dd

)

=a

b·(cf

df+

ed

fd

)=

a

b· cf + ed

df

=a(cf + ed)

bdf=

acf

bdf+

aed

bdf=

ac

bd+

ae

bf

=a

b· cd+

a

b· ef,

or the law of commutativity and law of associativity.

6 CHAPTER 2. FIELD (MATHEMATICS)

2.1.2 Second example: a field with four elements

In addition to familiar number systems such as the rationals, there are other, less immediate examples of fields. Thefollowing example is a field consisting of four elements called O, I, A and B. The notation is chosen such that O playsthe role of the additive identity element (denoted 0 in the axioms), and I is the multiplicative identity (denoted 1above). One can check that all field axioms are satisfied. For example:

A · (B + A) = A · I = A, which equals A · B + A · A = I + B = A, as required by the distributivity.

The above field is called a finite field with four elements, and can be denoted F4. Field theory is concerned withunderstanding the reasons for the existence of this field, defined in a fairly ad-hoc manner, and describing its innerstructure. For example, from a glance at the multiplication table, it can be seen that any non-zero element (i.e., I, A,and B) is a power of A: A = A1, B = A2 = A · A, and finally I = A3 = A · A · A. This is not a coincidence, but ratherone of the starting points of a deeper understanding of (finite) fields.

2.1.3 Alternative axiomatizations

As with other algebraic structures, there exist alternative axiomatizations. Because of the relations between the op-erations, one can alternatively axiomatize a field by explicitly assuming that there are four binary operations (add,subtract, multiply, divide) with axioms relating these, or (by functional decomposition) in terms of two binary oper-ations (add and multiply) and two unary operations (additive inverse and multiplicative inverse), or other variants.The usual axiomatization in terms of the two operations of addition and multiplication is brief and allows the otheroperations to be defined in terms of these basic ones, but in other contexts, such as topology and category theory, itis important to include all operations as explicitly given, rather than implicitly defined (compare topological group).This is because without further assumptions, the implicitly defined inverses may not be continuous (in topology), ormay not be able to be defined (in category theory). Defining an inverse requires that one is working with a set, not amore general object.For a very economical axiomatization of the field of real numbers, whose primitives are merely a set R with 1 ∈ R,addition, and a binary relation, "<". See Tarski’s axiomatization of the reals.

2.2 Related algebraic structures

The axioms imposed above resemble the ones familiar from other algebraic structures. For example, the existence ofthe binary operation "·", together with its commutativity, associativity, (multiplicative) identity element and inversesare precisely the axioms for an abelian group. In other words, for any field, the subset of nonzero elements F \ {0},also often denoted F×, is an abelian group (F×, ·) usually called multiplicative group of the field. Likewise (F, +) isan abelian group. The structure of a field is hence the same as specifying such two group structures (on the same set),obeying the distributivity.Important other algebraic structures such as rings arise when requiring only part of the above axioms. For example,if the requirement of commutativity of the multiplication operation · is dropped, one gets structures usually calleddivision rings or skew fields.

2.2.1 Remarks

By elementary group theory, applied to the abelian groups (F×, ·), and (F, +), the additive inverse −a and the multi-plicative inverse a−1 are uniquely determined by a.Similar direct consequences from the field axioms include

−(a · b) = (−a) · b = a · (−b), in particular −a = (−1) · a

as well as

a · 0 = 0.

Both can be shown by replacing b or c with 0 in the distributive property.

2.3. HISTORY 7

2.3 History

The concept of field was used implicitly by Niels Henrik Abel and Évariste Galois in their work on the solvability ofpolynomial equations with rational coefficients of degree five or higher.In 1857, Karl von Staudt published his Algebra of Throws which provided a geometric model satisfying the axiomsof a field.[2] This construction has been frequently recalled as a contribution to the foundations of mathematics.In 1871, Richard Dedekind introduced, for a set of real or complex numbers which is closed under the four arithmeticoperations, the German word Körper, which means “body” or “corpus” (to suggest an organically closed entity),[3]hence the common use of the letter K to denote a field. He also defined rings (then called order or order-modul), butthe term “a ring” (Zahlring) was invented by Hilbert.[4] In 1893, Eliakim Hastings Moore called the concept “field”in English.[5][6]

In 1881, Leopold Kronecker defined what he called a “domain of rationality”, which is indeed a field of polynomialsin modern terms. In 1893, Heinrich M. Weber gave the first clear definition of an abstract field.[7] In 1910, ErnstSteinitz published the very influential paper Algebraische Theorie der Körper (English: Algebraic Theory of Fields).[8]In this paper he axiomatically studies the properties of fields and defines many important field theoretic concepts likeprime field, perfect field and the transcendence degree of a field extension.Emil Artin developed the relationship between groups and fields in great detail from 1928 through 1942.

2.4 Examples

2.4.1 Rationals and algebraic numbers

The field of rational numbersQ has been introduced above. A related class of fields very important in number theoryare algebraic number fields. We will first give an example, namely the field Q(ζ) consisting of numbers of the form

a + bζ

with a, b ∈ Q, where ζ is a primitive third root of unity, i.e., a complex number satisfying ζ3 = 1, ζ ≠ 1. This fieldextension can be used to prove a special case of Fermat’s last theorem, which asserts the non-existence of rationalnonzero solutions to the equation

x3 + y3 = z3.

In the language of field extensions detailed below, Q(ζ) is a field extension of degree 2. Algebraic number fields areby definition finite field extensions of Q, that is, fields containing Q having finite dimension as a Q-vector space.

2.4.2 Reals, complex numbers, and p-adic numbers

Take the real numbersR, under the usual operations of addition and multiplication. When the real numbers are giventhe usual ordering, they form a complete ordered field; it is this structure which provides the foundation for mostformal treatments of calculus.The complex numbers C consist of expressions

a + bi

where i is the imaginary unit, i.e., a (non-real) number satisfying i2 = −1. Addition and multiplication of real numbersare defined in such a way that all field axioms hold for C. For example, the distributive law enforces

(a + bi)·(c + di) = ac + bci + adi + bdi2, which equals ac−bd + (bc + ad)i.

The real numbers can be constructed by completing the rational numbers, i.e., filling the “gaps": for example √2 issuch a gap. By a formally very similar procedure, another important class of fields, the field of p-adic numbers Qp isbuilt. It is used in number theory and p-adic analysis.

8 CHAPTER 2. FIELD (MATHEMATICS)

Hyperreal numbers and superreal numbers extend the real numbers with the addition of infinitesimal and infinitenumbers.

2.4.3 Constructible numbers

Given 0, 1, r1 and r2, the construction yields r1·r2

In antiquity, several geometric problems concerned the (in)feasibility of constructing certain numbers with compassand straightedge. For example, it was unknown to the Greeks that it is in general impossible to trisect a given angle.Using the field notion and field theory allows these problems to be settled. To do so, the field of constructible numbersis considered. It contains, on the plane, the points 0 and 1, and all complex numbers that can be constructed from thesetwo by a finite number of construction steps using only compass and straightedge. This set, endowed with the usualaddition and multiplication of complex numbers does form a field. For example, multiplying two (real) numbers r1and r2 that have already been constructed can be done using construction at the right, based on the intercept theorem.This way, the obtained field F contains all rational numbers, but is bigger than Q, because for any f ∈ F, the squareroot of f is also a constructible number.A closely related concept is that of a Euclidean field, namely an ordered field whose positive elements are closed undersquare root. The real constructible numbers form the least Euclidean field, and the Euclidean fields are precisely theordered extensions thereof.

2.4.4 Finite fields

Main article: Finite field

Finite fields (also called Galois fields) are fields with finitely many elements. The above introductory example F4 is afield with four elements. F2 consists of two elements, 0 and 1. This is the smallest field, because by definition a fieldhas at least two distinct elements 1 ≠ 0. Interpreting the addition and multiplication in this latter field as XOR andAND operations, this field finds applications in computer science, especially in cryptography and coding theory.In a finite field there is necessarily an integer n such that 1 + 1 + ··· + 1 (n repeated terms) equals 0. It can be shownthat the smallest such n must be a prime number, called the characteristic of the field. If a (necessarily infinite) field

2.5. SOME FIRST THEOREMS 9

has the property that 1 + 1 + ··· + 1 is never zero, for any number of summands, such as in Q, for example, thecharacteristic is said to be zero.A basic class of finite fields are the fields Fp with p elements (p a prime number):

Fp = Z/pZ = {0, 1, ..., p − 1},

where the operations are defined by performing the operation in the set of integers Z, dividing by p and taking theremainder; see modular arithmetic. A field K of characteristic p necessarily contains Fp,[9] and therefore may beviewed as a vector space over Fp, of finite dimension if K is finite. Thus a finite field K has prime power order, i.e., Khas q = pn elements (where n > 0 is the number of elements in a basis of K over Fp). By developing more field theory,in particular the notion of the splitting field of a polynomial f over a field K, which is the smallest field containing Kand all roots of f, one can show that two finite fields with the same number of elements are isomorphic, i.e., there isa one-to-one mapping of one field onto the other that preserves multiplication and addition. Thus we may speak ofthe finite field with q elements, usually denoted by Fq or GF(q).

2.4.5 Archimedean fields

Main article: Archimedean field

An Archimedean field is an ordered field such that for each element there exists a finite expression 1 + 1 + ··· + 1whose value is greater than that element, that is, there are no infinite elements. Equivalently, the field contains noinfinitesimals; or, the field is isomorphic to a subfield of the reals. A necessary condition for an ordered field tobe complete is that it be Archimedean, since in any non-Archimedean field there is neither a greatest infinitesimalnor a least positive rational, whence the sequence 1/2, 1/3, 1/4, …, every element of which is greater than everyinfinitesimal, has no limit. (And since every proper subfield of the reals also contains such gaps, up to isomorphismthe reals form the unique complete ordered field.)

2.4.6 Field of functions

Given a geometric object X, one can consider functions on such objects. Adding and multiplying them pointwise, i.e.,(f ⋅ g)(x) = f(x) ⋅ g(x) this leads to a field. However, for having multiplicative inverses, one has to consider partialfunctions, which, almost everywhere, are defined and have a non-zero value.If X is an algebraic variety over a field F, then the rational functions X → F form a field, the function field of X.This field consists of the functions that are defined and are the quotient of two polynomial functions outside somesubvariety. Likewise, if S is a Riemann surface, then the meromorphic functions S → C form a field. Under certaincircumstances, namely when S is compact, S can be reconstructed from this field.

2.4.7 Local and global fields

Another important distinction in the realm of fields, especially with regard to number theory, are local fields andglobal fields. Local fields are completions of global fields at a given place. For example, Q is a global field, andthe attached local fields are Qp and R (Ostrowski’s theorem). Algebraic number fields and function fields over Fqare further global fields. Studying arithmetic questions in global fields may sometimes be done by looking at thecorresponding questions locally—this technique is called local-global principle.

2.5 Some first theorems• Every finite subgroup of the multiplicative group F× is cyclic. This applies in particular to Fq×, it is cyclic oforder q − 1. In the introductory example, a generator of F4

× is the element A.

• A integral domain is a field if and only if it has no ideals except {0} and itself. Equivalently, an integral domainis a field if and only if its Krull dimension is 0.

• Isomorphism extension theorem

10 CHAPTER 2. FIELD (MATHEMATICS)

2.6 Constructing fields

2.6.1 Closure operations

Assuming the axiom of choice, for every field F, there exists a field F, called the algebraic closure of F, which containsF, is algebraic over F, which means that any element x of F satisfies a polynomial equation

fnxn + fn₋₁xn−1 + ··· + f1x + f0 = 0, with coefficients fn, ..., f0 ∈ F,

and is algebraically closed, i.e., any such polynomial does have at least one solution in F. The algebraic closure isunique up to isomorphism inducing the identity on F. However, in many circumstances in mathematics, it is notappropriate to treat F as being uniquely determined by F, since the isomorphism above is not itself unique. In thesecases, one refers to such a F as an algebraic closure of F. A similar concept is the separable closure, containing allroots of separable polynomials, instead of all polynomials.For example, if F =Q, the algebraic closureQ is also called field of algebraic numbers. The field of algebraic numbersis an example of an algebraically closed field of characteristic zero; as such it satisfies the same first-order sentencesas the field of complex numbers C.In general, all algebraic closures of a field are isomorphic. However, there is in general no preferable isomorphismbetween two closures. Likewise for separable closures.

2.6.2 Subfields and field extensions

A subfield is, informally, a small field contained in a bigger one. Formally, a subfield E of a field F is a subsetcontaining 0 and 1, closed under the operations +, −, · and multiplicative inverses and with its own operations definedby restriction. For example, the real numbers contain several interesting subfields: the real algebraic numbers, thecomputable numbers and the rational numbers are examples.The notion of field extension lies at the heart of field theory, and is crucial to many other algebraic domains. A fieldextension F / E is simply a field F and a subfield E ⊂ F. Constructing such a field extension F / E can be done by“adding new elements” or adjoining elements to the field E. For example, given a field E, the set F = E(X) of rationalfunctions, i.e., equivalence classes of expressions of the kind

p(X)

q(X),

where p(X) and q(X) are polynomials with coefficients in E, and q is not the zero polynomial, forms a field. This isthe simplest example of a transcendental extension of E. It also is an example of a domain (the ring of polynomialsE in this case) being embedded into its field of fractions E(X) .The ring of formal power series E[[X]] is also a domain, and again the (equivalence classes of) fractions of the formp(X)/ q(X) where p and q are elements of E[[X]] form the field of fractions for E[[X]] . This field is actually the ringof Laurent series over the field E, denoted E((X)) .In the above two cases, the added symbol X and its powers did not interact with elements of E. It is possible howeverthat the adjoined symbol may interact with E. This idea will be illustrated by adjoining an element to the field of realnumbersR. As explained above,C is an extension ofR.C can be obtained fromR by adjoining the imaginary symboli which satisfies i2 = −1. The result is that R[i]=C. This is different from adjoining the symbol X to R, because inthat case, the powers of X are all distinct objects, but here, i2=−1 is actually an element of R.Another way to view this last example is to note that i is a zero of the polynomial p(X) = X2 + 1. The quotientring R[X]/(X2 +1) can be mapped onto C using the map a+ bX → a+ ib . Since the ideal (X2+1) is generated by apolynomial irreducible over R, the ideal is maximal, hence the quotient ring is a field. This nonzero ring map fromthe quotient to C is necessarily an isomorphism of rings.The above construction generalises to any irreducible polynomial in the polynomial ring E[X], i.e., a polynomial p(X)that cannot be written as a product of non-constant polynomials. The quotient ring F = E[X] / (p(X)), is again a field.Alternatively, constructing such field extensions can also be done, if a bigger container is already given. Supposegiven a field E, and a field G containing E as a subfield, for example G could be the algebraic closure of E. Let x be

2.7. GALOIS THEORY 11

an element of G not in E. Then there is a smallest subfield of G containing E and x, denoted F = E(x) and called fieldextension F / E generated by x in G.[10] Such extensions are also called simple extensions. Many extensions are of thistype; see the primitive element theorem. For instance, Q(i) is the subfield of C consisting of all numbers of the forma + bi where both a and b are rational numbers.One distinguishes between extensions having various qualities. For example, an extension K of a field k is calledalgebraic, if every element of K is a root of some polynomial with coefficients in k. Otherwise, the extension is calledtranscendental. The aim of Galois theory is the study of algebraic extensions of a field.

2.6.3 Rings vs fields

Adding multiplicative inverses to an integral domain R yields the field of fractions of R. For example, the field offractions of the integers Z is just Q. Also, the field F(X) is the quotient field of the ring of polynomials F[X].Another method to obtain a field from a commutative ring R is taking the quotient R / m, where m is any maximalideal of R. The above construction of F = E[X] / (p(X)), is an example, because the irreducibility of the polynomialp(X) is equivalent to the maximality of the ideal generated by this polynomial. Another example are the finite fieldsFp = Z / pZ.

2.6.4 Ultraproducts

If I is an index set, U is an ultrafilter on I, and Fi is a field for every i in I, the ultraproduct of the Fi with respect toU is a field.For example, a non-principal ultraproduct of finite fields is a pseudo finite field; i.e., a PAC field having exactly oneextension of any degree.

2.7 Galois theory

Main article: Galois theory

Galois theory aims to study the algebraic extensions of a field by studying the symmetry in the arithmetic operations ofaddition and multiplication. The fundamental theorem of Galois theory shows that there is a strong relation betweenthe structure of the symmetry group and the set of algebraic extensions.In the case where F / E is a finite (Galois) extension, Galois theory studies the algebraic extensions of E that aresubfields of F. Such fields are called intermediate extensions. Specifically, the Galois group of F over E, denotedGal(F/E), is the group of field automorphisms of F that are trivial on E (i.e., the bijections σ : F → F that preserveaddition and multiplication and that send elements of E to themselves), and the fundamental theorem of Galois theorystates that there is a one-to-one correspondence between subgroups of Gal(F/E) and the set of intermediate extensionsof the extension F/E. The theorem, in fact, gives an explicit correspondence and further properties.To study all (separable) algebraic extensions of E at once, one must consider the absolute Galois group of E, definedas the Galois group of the separable closure, Esep, of E over E i.e., Gal(Esep/E). It is possible that the degree of thisextension is infinite (as in the case of E = Q). It is thus necessary to have a notion of Galois group for an infinitealgebraic extension. The Galois group in this case is obtained as a “limit” (specifically an inverse limit) of the Galoisgroups of the finite Galois extensions of E. In this way, it acquires a topology.[note 2] The fundamental theorem ofGalois theory can be generalized to the case of infinite Galois extensions by taking into consideration the topologyof the Galois group, and in the case of Esep/E it states that there this a one-to-one correspondence between closedsubgroups of Gal(Esep/E) and the set of all separable algebraic extensions of E (technically, one only obtains thoseseparable algebraic extensions of E that occur as subfields of the chosen separable closure Esep, but since all separableclosures of E are isomorphic, choosing a different separable closure would give the same Galois group and thus an“equivalent” set of algebraic extensions).

12 CHAPTER 2. FIELD (MATHEMATICS)

2.8 Generalizations

There are also proper classes with field structure, which are sometimes called Fields, with a capital F:

• The surreal numbers form a Field containing the reals, and would be a field except for the fact that they are aproper class, not a set.

• The nimbers form a Field. The set of nimbers with birthday smaller than 22n , the nimbers with birthday smallerthan any infinite cardinal are all examples of fields.

In a different direction, differential fields are fields equipped with a derivation. For example, the field R(X), togetherwith the standard derivative of polynomials forms a differential field. These fields are central to differential Galoistheory. Exponential fields, meanwhile, are fields equipped with an exponential function that provides a homomor-phism between the additive and multiplicative groups within the field. The usual exponential function makes the realand complex numbers exponential fields, denoted Rₑₓ and Cₑₓ respectively.Generalizing in a more categorical direction yields the field with one element and related objects.

2.8.1 Exponentiation

One does not in general study generalizations of fields with three binary operations. The familiar addition/subtraction,multiplication/division, exponentiation/root-extraction/logarithm operations from the natural numbers to the reals,each built up in terms of iteration of the last, mean that generalizing exponentiation as a binary operation is tempting,but has generally not proven fruitful; instead, an exponential field assumes a unary exponential function from theadditive group to the multiplicative group, not a partially defined binary function. Note that the exponential operationof ab is neither associative nor commutative, nor has a unique inverse ( ±2 are both square roots of 4, for instance),unlike addition and multiplication, and further is not defined for many pairs—for example, (−1)1/2 =

√−1 does not

define a single number. These all show that even for rational numbers exponentiation is not nearly as well-behaved asaddition and multiplication, which is why one does not in general axiomatize exponentiation.

2.9 Applications

The concept of a field is of use, for example, in defining vectors and matrices, two structures in linear algebra whosecomponents can be elements of an arbitrary field.Finite fields are used in number theory, Galois theory, cryptography, coding theory and combinatorics; and again thenotion of algebraic extension is an important tool.

2.10 See also

• Category of fields

• Glossary of field theory for more definitions in field theory.

• Heyting field

• Lefschetz principle

• Puiseux series

• Ring

• Vector space

• Vector spaces without fields

2.11. NOTES 13

2.11 Notes

[1] That is, the axiom for addition only assumes a binary operation +: F ×F → F, a, b 7→ a+ b. The axiom of inverse allows oneto define a unary operation − : F → F a 7→ −a that sends an element to its negative (its additive inverse); this is not takenas given, but is implicitly defined in terms of addition as " −a is the unique b such that a+ b = 0 ", “implicitly” because itis defined in terms of solving an equation—and one then defines the binary operation of subtraction, also denoted by "−",as − : F ×F → F, a, b 7→ a− b := a+ (−b) in terms of addition and additive inverse. In the same way, one defines the binaryoperation of division ÷ in terms of the assumed binary operation of multiplication and the implicitly defined operation of“reciprocal” (multiplicative inverse).

[2] As an inverse limit of finite discrete groups, it is equipped with the profinite topology, making it a profinite topologicalgroup

2.12 References

[1] Wallace, D A R (1998) Groups, Rings, and Fields, SUMS. Springer-Verlag: 151, Th. 2.

[2] Karl Georg Christian v. Staudt, Beiträge zur Geometrie der Lage (Contributions to the Geometry of Position), volume 2(Nürnberg, (Germany): Bauer and Raspe, 1857). See: “Summen vonWürfen” (sums of throws), pp. 166-171 ; “Produckteaus Würfen” (products of throws), pp. 171-176 ; “Potenzen von Würfen” (powers of throws), pp. 176-182.

[3] Peter Gustav Lejeune Dirichlet with R. Dedekind, Vorlesungen über Zahlentheorie von P. G. Lejeune Dirichlet (Lectureson Number Theory by P.G. Lejeune Dirichlet), 2nd ed., volume 1 (Braunschweig, Germany: Friedrich Vieweg und Sohn,1871), p. 424. From page 424: “Unter einem Körper wollen wir jedes System von unendlich vielen reellen oder complexenZahlen verstehen, welches in sich so abgeschlossen und vollständig ist, dass die Addition, Subtraction, Multiplication undDivision von je zwei dieser Zahlen immer wieder eine Zahl desselben Systems hervorbringt.” (By a “field” we will understandany system of infinitely many real or complex numbers, which is so closed and complete that the addition, subtraction,multiplication, and division of any two of these numbers always again produces a number of the same system.)

[4] J J O'Connor and E F Robertson, The development of Ring Theory, September 2004.

[5] Moore, E. Hastings (1893), “A doubly-infinite system of simple groups”, Bulletin of the New York Mathematical Society 3(3): 73–78, doi:10.1090/S0002-9904-1893-00178-X, JFM 25.0198.01. From page 75: “Such a system of s marks [i.e., afinite field with s elements] we call a field of order s.”

[6] Earliest Known Uses of Some of the Words of Mathematics (F)

[7] Fricke, Robert; Weber, Heinrich Martin (1924), Lehrbuch der Algebra, Vieweg, JFM 50.0042.03

[8] Steinitz, Ernst (1910), “Algebraische Theorie der Körper”, Journal für die reine und angewandte Mathematik 137: 167–309, doi:10.1515/crll.1910.137.167, ISSN 0075-4102, JFM 41.0445.03

[9] Jacobson (2009), p. 213

[10] Jacobson (2009), p. 213

2.13 Sources

• Artin, Michael (1991), Algebra, Prentice Hall, ISBN 978-0-13-004763-2, especially Chapter 13

• Allenby, R.B.J.T. (1991), Rings, Fields and Groups, Butterworth-Heinemann, ISBN 978-0-340-54440-2

• Blyth, T.S.; Robertson, E. F. (1985), Groups, rings and fields: Algebra through practice, Cambridge UniversityPress. See especially Book 3 (ISBN 0-521-27288-2) and Book 6 (ISBN 0-521-27291-2).

• Jacobson, Nathan (2009), Basic algebra 1 (2nd ed.), Dover, ISBN 978-0-486-47189-1

• James Ax (1968), The elementary theory of finite fields, Ann. of Math. (2), 88, 239–271

14 CHAPTER 2. FIELD (MATHEMATICS)

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

• Field Theory Q&A

• Fields at ProvenMath definition and basic properties.

• Field at PlanetMath.org.

Chapter 3

Glossary of field theory

Field theory is the branch of mathematics in which fields are studied. This is a glossary of some terms of the subject.(See field theory (physics) for the unrelated field theories in physics.)

3.1 Definition of a field

A field is a commutative ring (F,+,*) in which 0≠1 and every nonzero element has a multiplicative inverse. In a fieldwe thus can perform the operations addition, subtraction, multiplication, and division.The non-zero elements of a field F form an abelian group under multiplication; this group is typically denoted by F×;The ring of polynomials in the variable x with coefficients in F is denoted by F[x].

3.2 Basic definitionsCharacteristic The characteristic of the field F is the smallest positive integer n such that n·1 = 0; here n·1 stands

for n summands 1 + 1 + 1 + ... + 1. If no such n exists, we say the characteristic is zero. Every non-zerocharacteristic is a prime number. For example, the rational numbers, the real numbers and the p-adic numbershave characteristic 0, while the finite field Zp has characteristic p.

Subfield A subfield of a field F is a subset of F which is closed under the field operation + and * of F and which,with these operations, forms itself a field.

Prime field The prime field of the field F is the unique smallest subfield of F.

Extension field If F is a subfield of E then E is an extension field of F. We then also say that E/F is a field extension.

Degree of an extension Given an extension E/F, the field E can be considered as a vector space over the field F,and the dimension of this vector space is the degree of the extension, denoted by [E : F].

Finite extension A finite extension is a field extension whose degree is finite.

Algebraic extension If an element α of an extension field E over F is the root of a non-zero polynomial in F[x],then α is algebraic over F. If every element of E is algebraic over F, then E/F is an algebraic extension.

Generating set Given a field extension E/F and a subset S of E, we write F(S) for the smallest subfield of E thatcontains both F and S. It consists of all the elements of E that can be obtained by repeatedly using the operations+,−,*,/ on the elements of F and S. If E = F(S) we say that E is generated by S over F.

Primitive element An element α of an extension field E over a field F is called a primitive element if E=F(α), thesmallest extension field containing α. Such an extension is called a simple extension.

15

16 CHAPTER 3. GLOSSARY OF FIELD THEORY

Splitting field A field extension generated by the complete factorisation of a polynomial.

Normal extension A field extension generated by the complete factorisation of a set of polynomials.

Separable extension An extension generated by roots of separable polynomials.

Perfect field A field such that every finite extension is separable. All fields of characteristic zero, and all finite fields,are perfect.

Imperfect degree Let F be a field of characteristic p>0; then Fp is a subfield. The degree [F:Fp] is called theimperfect degree of F. The field F is perfect if and only if its imperfect degree is 1. For example, if F is afunction field of n variables over a finite field of characteristic p>0, then its imperfect degree is pn.[1]

Algebraically closed field A field F is algebraically closed if every polynomial in F[x] has a root in F; equivalently:every polynomial in F[x] is a product of linear factors.

Algebraic closure An algebraic closure of a field F is an algebraic extension of F which is algebraically closed.Every field has an algebraic closure, and it is unique up to an isomorphism that fixes F.

Transcendental Those elements of an extension field of F that are not algebraic over F are transcendental over F.

Algebraically independent elements Elements of an extension field of F are algebraically independent over F ifthey don't satisfy any non-zero polynomial equation with coefficients in F.

Transcendence degree The number of algebraically independent transcendental elements in a field extension. It isused to define the dimension of an algebraic variety.

3.3 Homomorphisms

Field homomorphism A field homomorphism between two fields E and F is a function

f : E → F

such that

f(x + y) = f(x) + f(y)

and

f(xy) = f(x) f(y)

for all x, y in E, as well as f(1) = 1. These properties imply that f(0) = 0, f(x−1) = f(x)−1 for x in E with x ≠ 0, andthat f is injective. Fields, together with these homomorphisms, form a category. Two fields E and F are calledisomorphic if there exists a bijective homomorphism

f : E → F.

The two fields are then identical for all practical purposes; however, not necessarily in a unique way. See, forexample, complex conjugation.

3.4 Types of fields

Finite field A field with finitely many elements.

Ordered field A field with a total order compatible with its operations.

3.5. FIELD EXTENSIONS 17

Rational numbers

Real numbers

Complex numbers

Number field Finite extension of the field of rational numbers.

Algebraic numbers The field of algebraic numbers is the smallest algebraically closed extension of the field ofrational numbers. Their detailed properties are studied in algebraic number theory.

Quadratic field A degree-two extension of the rational numbers.

Cyclotomic field An extension of the rational numbers generated by a root of unity.

Totally real field A number field generated by a root of a polynomial, having all its roots real numbers.

Formally real field

Real closed field

Global field A number field or a function field of one variable over a finite field.

Local field A completion of some global field (w.r.t. a prime of the integer ring).

Complete field A field complete w.r.t. to some valuation.

Pseudo algebraically closed field A field in which every variety has a rational point.[2]

Henselian field A field satisfying Hensel lemma w.r.t. some valuation. A generalization of complete fields.

Hilbertian field A field satisfying Hilbert’s irreducibility theorem: formally, one for which the projective line is notthin in the sense of Serre.[3][4]

Kroneckerian field A totally real algebraic number field or a totally imaginary quadratic extension of a totally realfield.[5]

CM-field or J-field An algebraic number field which is a totally imaginary quadratic extension of a totally realfield.[6]

Linked field A field over which no biquaternion algebra is a division algebra.[7]

Frobenius field A pseudo algebraically closed field whose absolute Galois group has the embedding property.[8]

3.5 Field extensions

Let E / F be a field extension.

Algebraic extension An extension in which every element of E is algebraic over F.

Simple extension An extension which is generated by a single element, called a primitive element, or generatingelement.[9] The primitive element theorem classifies such extensions.[10]

18 CHAPTER 3. GLOSSARY OF FIELD THEORY

Normal extension An extension that splits a family of polynomials: every root of the minimal polynomial of anelement of E over F is also in E.

Separable extension An algebraic extension in which the minimal polynomial of every element of E over F is aseparable polynomial, that is, has distinct roots.[11]

Galois extension A normal, separable field extension.

Primary extension An extension E/F such that the algebraic closure of F in E is purely inseparable over F; equiv-alently, E is linearly disjoint from the separable closure of F.[12]

Purely transcendental extension An extension E/F in which every element of E not in F is transcendental overF.[13][14]

Regular extension An extension E/F such that E is separable over F and F is algebraically closed in E.[12]

Simple radical extension A simple extension E/F generated by a single element α satisfying αn = b for an elementb of F. In characteristic p, we also take an extension by a root of an Artin–Schreier polynomial to be a simpleradical extension.[15]

Radical extension A tower F = F0 < F1 < · · · < Fk = E where each extension Fi/Fi−1 is a simple radicalextension.[15]

Self-regular extension An extension E/F such that E⊗FE is an integral domain.[16]

Totally transcendental extension An extension E/F such that F is algebraically closed in F.[14]

Distinguished class A class C of field extensions with the three properties[17]

1. If E is a C-extension of F and F is a C-extension of K then E is a C-extension of K.2. If E and F are C-extensions of K in a common overfieldM, then the compositum EF is a C-extension of

K.3. If E is a C-extension of F and E>K>F then E is a C-extension of K.

3.6 Galois theory

Galois extension A normal, separable field extension.

Galois group The automorphism group of a Galois extension. When it is a finite extension, this is a finite group oforder equal to the degree of the extension. Galois groups for infinite extensions are profinite groups.

Kummer theory The Galois theory of taking n-th roots, given enough roots of unity. It includes the general theoryof quadratic extensions.

Artin–Schreier theory Covers an exceptional case of Kummer theory, in characteristic p.

Normal basis A basis in the vector space sense of L over K, on which the Galois group of L over K acts transitively.

Tensor product of fields A different foundational piece of algebra, including the compositum operation (join offields).

3.7. EXTENSIONS OF GALOIS THEORY 19

3.7 Extensions of Galois theoryInverse problem of Galois theory Given a group G, find an extension of the rational number or other field with G

as Galois group.

Differential Galois theory The subject in which symmetry groups of differential equations are studied along thelines traditional in Galois theory. This is actually an old idea, and one of the motivations when Sophus Liefounded the theory of Lie groups. It has not, probably, reached definitive form.

Grothendieck’s Galois theory Avery abstract approach from algebraic geometry, introduced to study the analogueof the fundamental group.

3.8 References[1] Fried & Jarden (2008) p.45

[2] Fried & Jarden (2008) p.214

[3] Serre (1992) p.19

[4] Schinzel (2000) p.298

[5] Schinzel (2000) p.5

[6] Washington, Lawrence C. (1996). Introduction to Cyclotomic fields (2nd ed.). New York: Springer-Verlag. ISBN 0-387-94762-0. Zbl 0966.11047.

[7] Lam (2005) p.342

[8] Fried & Jarden (2008) p.564

[9] Roman (2007) p.46

[10] Lang (2002) p.243

[11] Fried & Jarden (2008) p.28

[12] Fried & Jarden (2008) p.44

[13] Roman (2007) p.102

[14] Isaacs, I. Martin (1994). Algebra: A Graduate Course. Graduate studies in mathematics 100. American MathematicalSociety. p. 389. ISBN 0-8218-4799-6. ISSN 1065-7339.

[15] Roman (2007) p.273

[16] Cohn, P. M. (2003). Basic Algebra. Groups, Rings, and Fields. Springer-Verlag. p. 427. ISBN 1-85233-587-4. Zbl1003.00001.

[17] Lang (2002) p.228

• Adamson, Iain T. (1982). Introduction to Field Theory (2nd ed.). Cambridge University Press. ISBN 0-521-28658-1.

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

• Lam, Tsit-Yuen (2005). Introduction to Quadratic Forms over Fields. Graduate Studies in Mathematics 67.American Mathematical Society. ISBN 0-8218-1095-2. MR 2104929. Zbl 1068.11023.

• Lang, Serge (1997). Survey ofDiophantine Geometry. Springer-Verlag. ISBN3-540-61223-8. Zbl 0869.11051.

• 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

20 CHAPTER 3. GLOSSARY OF FIELD THEORY

• Roman, Steven (2007). Field Theory. Graduate Texts in Mathematics 158. Springer-Verlag. ISBN 0-387-27678-5.

• Serre, Jean-Pierre (1989). Lectures on the Mordell-Weil Theorem. Aspects of Mathematics E15. Translatedand edited by Martin Brown from notes by Michel Waldschmidt. Braunschweig etc.: Friedr. Vieweg & Sohn.Zbl 0676.14005.

• Serre, Jean-Pierre (1992). Topics in Galois Theory. Research Notes in Mathematics 1. Jones and Bartlett.ISBN 0-86720-210-6. Zbl 0746.12001.

• Schinzel, Andrzej (2000). Polynomials with special regard to reducibility. Encyclopedia of Mathematics andIts Applications 77. Cambridge: Cambridge University Press. ISBN 0-521-66225-7. Zbl 0956.12001.

Chapter 4

Heyting field

A Heyting field is one of the inequivalent ways in constructive mathematics to capture the classical notion of a field.It is essentially a field with an apartness relation. The key field axiom is that an element is invertible if and only if itis not zero. In a Heyting field, this is taken to mean that it is apart from zero. In many cases, the assumption that anelement is not equal to zero is insufficient to construct the inverse; the assumption that it is apart from zero implicitlycontains the necessary information.The prototypical Heyting field is the real numbers with their natural apartness relation.

4.1 References• Mines, Richman, Ruitenberg. A Course in Constructive Algebra. Springer, 1987.

21

22 CHAPTER 4. HEYTING FIELD

4.2 Text and image sources, contributors, and licenses

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