Integrally closed domain

In commutative algebra, an integrally closed domain A is an integral domain whose integral closure in its field of fractions is A itself. Many well-studied domains are integrally closed: Fields, the ring of integers Z, unique factorization domains and regular local rings are all integrally closed. In commutative algebra, an integrally closed domain A is an integral domain whose integral closure in its field of fractions is A itself. Many well-studied domains are integrally closed: Fields, the ring of integers Z, unique factorization domains and regular local rings are all integrally closed. To give a non-example, let k be a field and A = k [ t 2 , t 3 ] ⊂ B = k [ t ] {displaystyle A=ksubset B=k} (A is the subalgebra generated by t2 and t3.) A and B have the same field of fractions, and B is the integral closure of A (since B is a UFD.) In other words, A is not integrally closed. This is related to the fact that the plane curve Y 2 = X 3 {displaystyle Y^{2}=X^{3}} has a singularity at the origin. Note that integrally closed domains appear in the following chain of class inclusions: Let A be an integrally closed domain with field of fractions K and let L be a finite extension of K. Then x in L is integral over A if and only if its minimal polynomial over K has coefficients in A. This implies in particular that an integral element over an integrally closed domain A has a minimal polynomial over A. This is stronger than the statement that any integral element satisfies some monic polynomial. In fact, the statement is false without 'integrally closed'. (For example, consider A = Z [ 5 ] {displaystyle A=mathbb {Z} } , which is not integrally closed over Z {displaystyle mathbb {Z} } because it does not for example contain the element 5 + 1 2 {displaystyle {frac {{sqrt {5}}+1}{2}}} of its field of fractions, which satisfies the monic integral polynomial X 2 − X − 1 = 0 {displaystyle X^{2}-X-1=0} ). 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.