Field theory glossary
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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 nonzero 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
 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 nonzero characteristic is a prime number. For example, the rational numbers, the real numbers and the padic 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
 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 nonzero 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
 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
 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 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 p^{n}.
 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 essentially unique.
 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 polynomial equation with coefficients in F.
 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
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, 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
 A field with finitely many elements.
 Ordered field
 A field with a total order compatible with its operations.
 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 of rational numbers. Their detailed properties are studied in algebraic number theory.
 Quadratic field
 A degreetwo 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).
 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 of order equal to the degree of the extension. Galois groups for infinite extensions are profinite groups.
 Kummer theory
 The Galois theory of taking nth roots, given enough roots of unity. It includes the general theory of quadratic extensions.
 ArtinSchreier 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 of fields).
Extensions 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
 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
 A very abstract approach from algebraic geometry, introduced to study the analogue of the fundamental group.
Published  March 2009
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