What is "algebraic integer"

n. (context mathematics English) A number (real or complex) which is a root of a monic polynomial whose coefficients are integers.

In number theory, an algebraic integer is a complex number that is a root of some monic polynomial (a polynomial whose leading coefficient is 1) with coefficients in (the set of integers). The set of all algebraic integers is closed under addition and multiplication and therefore is a subring of complex numbers denoted by A. The ring A is the integral closure of regular integers in complex numbers.

The ring of integers of a number field K, denoted by O, is the intersection of K and A: it can also be characterised as the maximal order of the field K. Each algebraic integer belongs to the ring of integers of some number field. A number x is an algebraic integer if and only if the ring [x] is finitely generated as an abelian group, which is to say, as a -module.