algebraic number field

If r is an algebraic number of degree n, then the totality of all expressions that can be constructed from r by repeated additions, subtractions, multiplications, and divisions is called a number field (or an algebraic number field) generated by r, and is denoted F[r]. Formally, a number field is a finite extension Q(α) of the field Q of rational numbers. The elements of a number field which are roots of a polynomial z^n + a_(n - 1) z^(n - 1) + ... + a_0 = 0 with integer coefficients and leading coefficient 1 are called the algebraic integers of that field.

algebraic integer | algebraic number | field | finite field | function field | local field | number field order | number field sieve | number field signature | number ring | Q | quadratic field