In commutative algebra, an integrally closed domain A is an integral domain whose integral closure in its field of fractions is A itself. Spelled out, this means that if x is an element of the field of fractions of A that is a root of a monic polynomial with coefficients in A, then x is itself an element of A. Many well-studied domains are integrally closed, as shown by the following chain of class inclusions:

An explicit example is the ring of integers Z, a Euclidean domain. All regular local rings are integrally closed as well.

A ring whose localizations at all prime ideals are integrally closed domains is a normal ring.

Let A be an integrally closed domain with field of fractions K and let L be a field extension of K. Then x∈L is integral over A if and only if it is algebraic over K and its minimal polynomial over K has coefficients in A. In particular, this means that any element of L integral over A is root of a monic polynomial in A[X] that is irreducible in K[X].

If A is a domain contained in a field K, we can consider the integral closure of A in K (i.e. the set of all elements of K that are integral over A). This integral closure is an integrally closed domain.

Integrally closed domains also play a role in the hypothesis of the Going-down theorem. The theorem states that if A⊆B is an integral extension of domains and A is an integrally closed domain, then the going-down property holds for the extension A⊆B.

The following are integrally closed domains.

To give a non-example, let k be a field and ${\displaystyle A=k[t^{2},t^{3}]\subset k[t]}$, the subalgebra generated by t² and t³. Then A is not integrally closed: it has the field of fractions ${\displaystyle k(t)}$, and the monic polynomial ${\displaystyle X^{2}-t^{2}}$ in the variable X has root t which is in the field of fractions but not in A. This is related to the fact that the plane curve ${\displaystyle Y^{2}=X^{3}}$ has a singularity at the origin.