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The algebraic integers in a number field.

An element of an extension field of a field F which is not algebraic over F. A transcendental number is a complex number which is transcendental over the field Q of rational ...

A unimodular matrix is a real square matrix A with determinant det(A)=+/-1 (Born and Wolf 1980, p. 55; Goldstein 1980, p. 149). More generally, a matrix A with elements in ...

A polynomial with coefficients in a field is separable if its factors have distinct roots in some extension field.

Let (K,|·|) be a non-Archimedean field. Its valuation ring R is defined to be R={x in K:|x|<=1}. The valuation ring has maximal ideal M={x in K:|x|<1}, and the field R/M is ...

The field F^_ is called an algebraic closure of F if F^_ is algebraic over F and if every polynomial f(x) in F[x] splits completely over F^_, so that F^_ can be said to ...

A theorem which treats constructions of fields of field characteristic p.

A field K is said to be algebraically closed if every polynomial with coefficients in K has a root in K.

An extension A subset B of a group, ring, module, field, etc., such that A!=B.

The doublestruck letter R denotes the field of real numbers.