# nLab the fundamental theorem of Galois theory

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An instance is the following:

Let $k$ be a field, let ${k}_{s}$ denote the separable closure of $k$.

###### Theorem

The functor

$\left\{\begin{array}{l}{\mathrm{Sch}}_{\mathrm{et}}\to \mathrm{Gal}\left({k}_{s}/k\right)-\mathrm{Mod}\\ X↦X\left({k}_{s}\right)\end{array}$\begin{cases} Sch_{et}\to Gal(k_s / k)- Mod \\ X\mapsto X(k_s) \end{cases}

from étale schemes to the category of Galois modules $\mathrm{Gal}\left({k}_{s}/s\right)-\mathrm{Mod}$ is an equivalence of categories.

Revised on June 7, 2012 19:44:50 by Stephan Alexander Spahn (79.227.183.2)