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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,〔Taken from Matsumura〕 let ''k'' be a field and (''A'' is the subalgebra generated by ''t''2 and ''t''3.) ''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 has a singularity at the origin. 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''.〔Matsumura, Theorem 9.2〕 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" (consider ) 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''. Note that integrally closed domains appear in the following chain of class inclusions: == Examples == The following are integrally closed domains. *Any principal ideal domain (in particular, any field). *Any unique factorization domain (in particular, any polynomial ring over a unique factorization domain.) *Any GCD domain (in particular, any Bézout domain or valuation domain). *Any Dedekind domain. *Any symmetric algebra over a field (since every symmetric algebra is isomorphic to a polynomial ring in several variables over a field). 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「integrally closed domain」の詳細全文を読む スポンサード リンク
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