existence of extensions of field isomorphisms to splitting fields

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

Theorem.

Let $\sigma:F\rightarrow F^{\prime}$ be an isomorphism of fields, $S=\{f_{\alpha}:\alpha\in A\}$ a set of non-constant polynomials in $F[X]$, and $S^{\prime}=\{\sigma(f_{\alpha}):\alpha\in A\}$ 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.

Title existence of extensions of field isomorphisms to splitting fields ExistenceOfExtensionsOfFieldIsomorphismsToSplittingFields 2013-03-22 18:37:59 2013-03-22 18:37:59 azdbacks4234 (14155) azdbacks4234 (14155) 4 azdbacks4234 (14155) Theorem msc 12F05 SplittingField