Integral element

In commutative algebra, an element b of a commutative ring B is said to be integral over a subring A of B if b is a root of some monic polynomial over A.^[1]

If A, B are fields, then the notions of "integral over" and of an "integral extension" are precisely "algebraic over" and "algebraic extensions" in field theory (since the root of any polynomial is the root of a monic polynomial).

The case of greatest interest in number theory is that of complex numbers integral over Z (e.g., ${\displaystyle {\sqrt {2}}}$ or ${\displaystyle 1+i}$); in this context, the integral elements are usually called algebraic integers. The algebraic integers in a finite extension field k of the rationals Q form a subring of k, called the ring of integers of k, a central object of study in algebraic number theory.

In this article, the term ring will be understood to mean commutative ring with a multiplicative identity.