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existence of extensions of field isomorphisms to splitting fields
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(Theorem)
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The following theorem implies the essential uniqueness of splitting fields and algebraic closures.
Theorem Let $\sigma:F\rightarrow F^\prime$ be an isomorphism of fields, $S=\set{f_\alpha:\alpha\in A}$ a set of non-constant polynomials in $F[X]$ , and $S^\prime=\set{\sigma(f_\alpha):\alpha\in A}$ the corresponding set of polynomials in $F^\prime[X]$ . If $K$ is a splitting field of $S$ over $F$ and $K^\prime$ a splitting field of $S^\prime$ over $F^\prime$ , then $\sigma$ may be extended to an isomorphism of $K$ and $K^\prime$ .
Corollary If $F$ is a field and $S$ a set of non-constant polynomials in $F[X]$ , then any two splitting fields of $S$ over $F$ are $F$ -isomorphic. In particular, any two algebraic closures of $F$ are $F$ -isomorphic.
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"existence of extensions of field isomorphisms to splitting fields" is owned by azdbacks4234.
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Cross-references: polynomials, fields, isomorphism, algebraic closures, splitting fields, implies, theorem
This is version 1 of existence of extensions of field isomorphisms to splitting fields, born on 2008-12-22.
Object id is 11372, canonical name is ExistenceOfExtensionsOfFieldIsomorphismsToSplittingFields.
Accessed 333 times total.
Classification:
| AMS MSC: | 12F05 (Field theory and polynomials :: Field extensions :: Algebraic extensions) |
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Pending Errata and Addenda
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