Glossary of field theory

Definition of a field

A field is an commutative ring (F,+,*) of which every nonzero element is invertible. Over a field, we can perform addition, subtraction, multiplication and division.

The abelian group of non-zero elements of a field F is typically denoted by F^×;

;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.

The ring of polynomials with coefficients in F is denoted by F[x].

Basic definitions

; 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 : A prime field is the unique smallest subfield of F.

; Extension field : If F is a subfield of E then E is an extension field of F.

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

; Primitive element : A 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.

; Algebraically closed field : The largest unique algebraic extension field of F.

; Transcendental : If an element is not algebraic over F, then it is transcendental.

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.

Types of fields

; Finite field : A field of finitely many elements.

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

; Rational numbers

; Real numbers

; Complex numbers

; Number field : Algebraic extension of the field of rational numbers.

; Algebraic numbers : The field of algebraic numbers is the algebraically closed extension of the field of rational numbers.

Referenced By