Integral Element

Integral Element

In commutative algebra, an element of a commutative ring is said to be integral over, a subring of, if there is an and such that

That is to say, is a root of a monic polynomial over . If every element of B is integral over A, then it is said that B is integral over A, or equivalently B is an integral extension of A.

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 special case of greatest interest in number theory is that of complex numbers integral over Z; in this context, they are usually called algebraic integers (e.g., ). A ring consisting of the algebraic integers of a finite extension field of the rationals is called the ring of integers of, a central object in algebraic number theory.

The set of elements of B that are integral over A is called the integral closure of A in B. It is a subring of B containing A.

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