nLab integral closure

Definition

Given a commutative ring $k$ and a field $L\supset k$, an element $x\in L$ is said to be integral over $k$ if it satisfies a monic polynomial equation with coefficients in $k$, or equivalently, there exist a finitely-generated nonzero $k$-submodule $M\subset L$ such that $x M \subset M$.

A commutative ring $K\supset k$ is said to be integral over $k$ if every element of $K$ is integral over $k$. The relation of integrality of overrings is transitive. If $f:K\to K'$ is a surjective homomorphism of rings and $K$ integral over $k\subset K$, then $K' = f(K)$ is integral over $f(k)$.

The set of all elements of $L$ integral over $k$ is a subring of $L$ called the integral closure of $k$ in $L$. We say that $k$ is integrally closed in $L$ if it equals its own integral closure in $L$.

An integral domain $k$ is integrally closed if it is integrally closed in the quotient field of $k$.

Properties

If $k$ is an integrally closed Noetherian domain and $L$ a finite separable field extension of its quotient field $Q(k)$ then the integral closure of $k$ in $L$ is finitely generated over $k$.

If $k$ is a principal ideal ring and $L$ a finite separable extension of degree $n$ of its quotient field $Q(k)$, then the integral closure of $k$ in $L$ is a free rank $n$-module over $k$.

If $K$ is integral over a subring $k$ then for any multiplicative set $S\subset k$, the localization $S^{-1} K$ is integral over $S^{-1} k$.

Every unique factorization domain is integrally closed.

In constructive mathematics

In constructive mathematics, integral closure and algebraic closure are not the same if the field is not a discrete field. Integral closure is usually better behaved than algebraic closure in fields which are not discrete fields, because not every polynomial can be shown to have a well-defined degree, but every monic polynomial by definition has a well-defined degree. This is especially the case with the fundamental theorem of algebra for a set of complex numbers, which in constructive mathematics states that the complex numbers are integrally closed, and is true for the modulated Cantor real numbers while not provable for the Dedekind real numbers.

See also

• fundamental theorem of algebra

 References

• Serge Lang, Algebraic number theory, GTM 110, Springer 1970, 2000
• O. Zariski, Samuel, Commutative algebra
• N. Bourbaki, Commutative algebra
• Z. Borevich, I. Shafarevich, Number theory
• E. Artin, J. Tate, Class field theory, 1967
• A. Weil, Basic number theory