Algebraic Integer

An algebraic integer is a root of a monic polynomial with integer coefficients — a number that is integral over the ring of ordinary integers Z. The set of algebraic integers in a number field forms a ring, the ring of integers, which is the natural setting for arithmetic in that field. Unlike the field itself, the ring of integers is a Dedekind domain, in which every nonzero ideal factors uniquely into prime ideals.

Algebraic integers bridge elementary number theory and commutative algebra: they are the elements that make the arithmetic of a number field structurally tractable. The study of their factorization properties, units, and ideal structure is the foundation of algebraic number theory.

The term algebraic integer is sometimes mistaken for a generalization of the ordinary integers. The opposite is true. The ordinary integers are the special case. The algebraic integers reveal what integer-ness actually is: not a property of the number line, but a structural property of integrality in any ring.