Algebraic integers

In Algebraic 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 ${\displaystyle \mathbb {Z} }$ (the set of integers). The set of all algebraic integers, A, is closed under addition and multiplication and therefore is a commutative subring of the complex numbers. The ring A is the integral closure of regular integers ${\displaystyle \mathbb {Z} }$ in complex numbers.

The ring of integers of a number field K, denoted by O_K, 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 ${\displaystyle \mathbb {Z} [x]}$ is finitely generated as an abelian group, which is to say, as a ${\displaystyle \mathbb {Z} }$-module.

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