Glossary of field theory

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.) 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]. 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 Zp where p is 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.