That is, an algebraic integer is a complex root of some monic polynomial (a polynomial whose leading coefficient is 1) whose coefficients are integers.
The set of all algebraic integers A is closed under addition, subtraction and multiplication and therefore is a commutative subring of the complex numbers.
The ring of integers of a number field K, denoted by OK, 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 α is an algebraic integer if and only if the ring
Algebraic integers are a special case of integral elements of a ring extension.
In particular, an algebraic integer is an integral element of a finite extension
, is finitely generated if and only if α is an algebraic integer.
The proof is analogous to that of the corresponding fact regarding algebraic numbers, with
here, and the notion of field extension degree replaced by finite generation (using the fact that
is finitely generated itself); the only required change is that only non-negative powers of α are involved in the proof.
This can be shown analogously to the corresponding proof for algebraic numbers, using the integers
One may also construct explicitly the monic polynomial involved, which is generally of higher degree than those of the original algebraic integers, by taking resultants and factoring.
In other words, the algebraic integers form a ring that is integrally closed in any of its extensions.