Colloquially, an algebraic integer is a solution to an equation
where each is an integer. More precisely, an element belonging to an algebraic extension of is an (algebraic) integer, or more briefly is integral, if it satisfies an equation of the form (1). Equivalently, if is an algebraic extension of (e.g., if is a number field), an element is integral if the subring is finitely generated as a -module.
If are integral over (say), then and are integral over . For, if is integral over , it is a fortiori integral over , hence
is finitely generated over and therefore, since is integral, also finitely generated over . It follows that the submodules and are therefore also finitely generated over (since is a Noetherian ring). Thus the integral elements form a ring. In particular, the integral elements in a number field form a ring often denoted by , usually called the ring of integers in . This ring is a Dedekind domain?.