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 in which 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 The characteristic of the field F is the smallest positive integer n such that ; here n·1 stands for n summands . 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 Zp with p being prime 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 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 , 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 ; then Fp is a subfield. The degree 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 , then its imperfect degree is pn. 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 if they 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 is used to define the dimension of an algebraic variety.
Homomorphisms
Field homomorphism A field homomorphism between two fields E and F is a ring homomorphism, i.e., 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. For fields E and F, these properties imply that , for x in E×, 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 conjugate.