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- Integrally_closed_domain abstract "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 a field). 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. 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 domain appear in the following chain of class inclusions: Commutative rings ⊃ integral domains ⊃ integrally closed domains ⊃ unique factorization domains ⊃ principal ideal domains ⊃ Euclidean domains ⊃ fields↑ ↑".
- Integrally_closed_domain wikiPageID "21420209".
- Integrally_closed_domain wikiPageRevisionID "606793875".
- Integrally_closed_domain hasPhotoCollection Integrally_closed_domain.
- Integrally_closed_domain subject Category:Commutative_algebra.
- Integrally_closed_domain comment "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 a field). 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.".
- Integrally_closed_domain label "Domínio integralmente fechado".
- Integrally_closed_domain label "Integrally closed domain".
- Integrally_closed_domain sameAs Celistvě_uzavřený_obor.
- Integrally_closed_domain sameAs Domínio_integralmente_fechado.
- Integrally_closed_domain sameAs m.0h3tcfz.
- Integrally_closed_domain sameAs Q6042757.
- Integrally_closed_domain sameAs Q6042757.
- Integrally_closed_domain wasDerivedFrom Integrally_closed_domain?oldid=606793875.
- Integrally_closed_domain isPrimaryTopicOf Integrally_closed_domain.