# 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 |
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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 |