1. Definition of a field

2. Basic definitions

3. Homomorphisms

4. Types of fields

5. Field extensions

6. Galois theory

7. Extensions of Galois theory

8. References

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.)

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 field we 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].

Basic definitions

//Characteristic (algebra)">Characteristic : 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-zero characteristic is a prime number. For example, the rational numbers, the real numbers and the p-adic numbers have characteristic 0, while the finite field Z_p 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">Prime field : The prime field of the field F is the unique smallest subfield of F.

//Field extension">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 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">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 that contains 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 (field theory)">Primitive element : An element α of an extension field E over a field F is called a primitive element if E=F(α), the smallest extension field containing α. Such an extension is called a simple extension.

//Splitting field">Splitting field : A field extension generated by the complete factorisation of a polynomial.

//Normal extension">Normal extension : A field extension generated by the complete factorisation of a set of polynomials.

//Separable extension">Separable extension : An extension generated by roots of separable polynomials.

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

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

//Algebraically closed field">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">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 element">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 if they don't satisfy any non-zero polynomial equation with coefficients in F.

//Transcendence degree">Transcendence degree : The number of algebraically independent transcendental elements in a field extension. It is used to define the dimension of an algebraic variety.

Homomorphisms

Field homomorphism

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

f : E → F

such that, for all x, y in E,

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

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

f(1) = 1.

These properties imply that {{nowrap|1=f(0) = 0}}, {{nowrap|1=f(x⁻¹) = f(x)⁻¹}} for x in E with {{nowrap|x ≠ 0}}, and that f is injective. Fields, together with these homomorphisms, form a category. Two fields E and F are called isomorphic 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, for example, complex conjugation.

Types of fields

//Finite field">Finite field : A field with finitely many elements. Aka Galois field.

//Ordered field">Ordered field : A field with a total order compatible with its operations.

//Rational number">Rational numbers

//Real number">Real numbers

//Complex number">Complex numbers

//Number field">Number field : Finite extension of the field of rational numbers.

//Algebraic number">Algebraic numbers : The field of algebraic numbers is the smallest algebraically closed extension of the field of rational numbers. Their detailed properties are studied in algebraic number theory.

//Quadratic field">Quadratic field : A degree-two extension of the rational numbers.

//Cyclotomic field">Cyclotomic field : An extension of the rational numbers generated by a root of unity.

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

//Formally real field">Formally real field

//Real closed field">Real closed field

//Global field">Global field : A number field or a function field of one variable over a finite field.

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

//Complete field">Complete field : A field complete w.r.t. to some valuation.

//Pseudo algebraically closed field">Pseudo algebraically closed field : A field in which every variety has a rational point.^[2]

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

//Hilbertian field">Hilbertian field: A field satisfying Hilbert's irreducibility theorem: formally, one for which the projective line is not thin in the sense of Serre.^[3][4]

Kroneckerian field

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

//CM-field">CM-field or J-field: An algebraic number field which is a totally imaginary quadratic extension of a totally real field.^[6]

//Linked field">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]

Field extensions

Let E / F be a field extension.

//Algebraic extension">Algebraic extension : An extension in which every element of E is algebraic over F.

//Simple extension">Simple extension: An extension which is generated by a single element, called a primitive element, or generating element.^[9] The primitive element theorem classifies such extensions.^[10]

//Normal extension">Normal extension : An extension that splits a family of polynomials: every root of the minimal polynomial of an element of E over F is also in E.

//Separable extension">Separable extension : An algebraic extension in which the minimal polynomial of every element of E over F is a separable polynomial, that is, has distinct roots.^[11]

//Galois extension">Galois extension : A normal, separable field extension.

//Primary extension">Primary extension : An extension E/F such that the algebraic closure of F in E is purely inseparable over F; equivalently, E is linearly disjoint from the separable closure of F.^[12]

//Purely transcendental extension">Purely transcendental extension : An extension E/F in which every element of E not in F is transcendental over F.^[13][14]

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

//Simple radical extension">Simple radical extension: A simple extension E/F generated by a single element α satisfying for an element b of F. In characteristic p, we also take an extension by a root of an Artin–Schreier polynomial to be a simple radical extension.^[14]

//Radical extension">Radical extension: A tower where each extension is a simple radical extension.^[14]

//Self-regular extension">Self-regular extension : An extension E/F such that E⊗_FE is an integral domain.^[15]

Totally transcendental extension

An extension E/F such that F is algebraically closed in F.^[16]

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 overfield M, 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.

Galois theory

//Galois extension">Galois extension : A normal, separable field extension.

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

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

//Artin–Schreier theory">Artin–Schreier theory : Covers an exceptional case of Kummer theory, in characteristic p.

//Normal basis">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">Tensor product of fields : A different foundational piece of algebra, including the compositum operation (join of fields).

Extensions of Galois theory

//Inverse problem of Galois theory">Inverse 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">Differential Galois theory : The subject in which symmetry groups of differential equations are studied along the lines traditional in Galois theory. This is actually an old idea, and one of the motivations when Sophus Lie founded the theory of Lie groups. It has not, probably, reached definitive form.

//Grothendieck's Galois theory">Grothendieck's Galois theory : A very abstract approach from algebraic geometry, introduced to study the analogue of the fundamental group.

References

• {{cite book | title=Introduction to Field Theory | first=Iain T. | last=Adamson | edition=2nd | publisher=Cambridge University Press | year=1982 | isbn=0-521-28658-1 }}
• {{cite book | last1=Fried | first1=Michael D. | last2=Jarden | first2=Moshe | title=Field arithmetic | edition=3rd revised | series=Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge | volume=11 | publisher=Springer-Verlag | year=2008 | isbn=978-3-540-77269-9 | zbl=1145.12001 }}
• {{cite book | title=Introduction to Quadratic Forms over Fields | volume=67 | series=Graduate Studies in Mathematics | first=Tsit-Yuen | last=Lam | author-link=T. Y. Lam | publisher=American Mathematical Society | year=2005 | isbn=0-8218-1095-2 | zbl=1068.11023 | mr = 2104929 }}
• {{cite book | first=Serge | last=Lang | authorlink=Serge Lang | title=Survey of Diophantine Geometry | publisher=Springer-Verlag | year=1997 | isbn=3-540-61223-8 | zbl=0869.11051 }}
• {{Lang Algebra | edition=3r}}
• {{cite book | title=Field Theory | volume=158 | series=Graduate Texts in Mathematics | first=Steven | last=Roman | publisher=Springer-Verlag | year=2007 | isbn=0-387-27678-5 }}
• {{cite book | first=Jean-Pierre | last=Serre | authorlink=Jean-Pierre Serre | title=Lectures on the Mordell-Weil Theorem | year=1989 | others=Translated and edited by Martin Brown from notes by Michel Waldschmidt | zbl=0676.14005 | series=Aspects of Mathematics | volume=E15 | location=Braunschweig etc. | publisher=Friedr. Vieweg & Sohn }}
• {{cite book | first=Jean-Pierre | last=Serre | authorlink=Jean-Pierre Serre | title=Topics in Galois Theory | series=Research Notes in Mathematics | volume=1 | publisher=Jones and Bartlett | year=1992 | isbn=0-86720-210-6 | zbl=0746.12001 }}
• {{cite book | last=Schinzel | first= Andrzej | authorlink=Andrzej Schinzel | title=Polynomials with special regard to reducibility | zbl=0956.12001 | series=Encyclopedia of Mathematics and Its Applications | volume=77 | location=Cambridge | publisher=Cambridge University Press | year=2000 | isbn=0-521-66225-7 }}

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