nLab integral closure



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

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

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

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


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

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

If KK is integral over a subring kk then for any multiplicative set SkS\subset k, the localization S 1KS^{-1} K is integral over S 1kS^{-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


  • 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

Last revised on December 10, 2022 at 14:25:45. See the history of this page for a list of all contributions to it.