existence of extensions of field isomorphisms to splitting fields

The following theorem implies the essential uniqueness of splitting fields and algebraic closures.

Theorem.

Let $\sigma \mathrm{:}F\mathrm{\to }{F}^{\mathrm{\prime }}$ be an isomorphism of fields, $S\mathrm{=}\mathrm{\{}{f}_{\alpha }\mathrm{:}\alpha \mathrm{\in }A\mathrm{\}}$ a set of non-constant polynomials in $F\mathit{}\mathrm{[}X\mathrm{]}$, and $S^{\mathrm{\prime }}\mathrm{=}\mathrm{\{}\sigma \mathit{}\mathrm{(}{f}_{\alpha }\mathrm{)}\mathrm{:}\alpha \mathrm{\in }A\mathrm{\}}$ the corresponding set of polynomials in $F^{\mathrm{\prime }}\mathit{}\mathrm{[}X\mathrm{]}$. If $K$ is a splitting field of $S$ over $F$ and $K^{\mathrm{\prime }}$ a splitting field of $S^{\mathrm{\prime }}$ over $F^{\mathrm{\prime }}$, then $\sigma$ may be extended to an isomorphism of $K$ and $K^{\mathrm{\prime }}$.

Corollary.

If $F$ is a field and $S$ a set of non-constant polynomials in $F\mathit{}\mathrm{[}X\mathrm{]}$, then any two splitting fields of $S$ over $F$ are $F$-isomorphic. In particular, any two algebraic closures of $F$ are $F$-isomorphic.

Title	existence of extensions of field isomorphisms to splitting fields
Canonical name	ExistenceOfExtensionsOfFieldIsomorphismsToSplittingFields
Date of creation	2013-03-22 18:37:59
Last modified on	2013-03-22 18:37:59
Owner	azdbacks4234 (14155)
Last modified by	azdbacks4234 (14155)
Numerical id	4
Author	azdbacks4234 (14155)
Entry type	Theorem
Classification	msc 12F05
Related topic	SplittingField