Glossary of Field Theory - Basic Definitions

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 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

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 .

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, 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 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 p>0, then its imperfect degree is pn.

Algebraically closed field

A field F is algebraically closed if every polynomial in F has a root in F; equivalently: every polynomial in F 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.