## Galois connections ## Galois theory of fields ## Galois theory of schemes +-- {: .num_theorem} ###### Theorem ([[the fundamental theorem of Galois theory|fundamental theorem of Galois theory]]) Let $k$ be a field, let $k_s$ denote its separable closure. The functor $$\begin{cases} k.Sch_{et}\to Gal(k_s / k)- Mod \\ X\mapsto X(k_s) \end{cases}$$ from [[etale scheme|étale k-schemes]] to the category of [[Galois module|Galois modules]] $Gal(k_s/s)-Mod$ is an equivalence of categories. Here $Gal (k_s/k)$ is considered as a profinite topological group. [Demazure, section I.8, p.17](#Dem) =-- ## Grothendieck's Galois theory ## Galois descend ## References * Demazure, [[Lectures on p-divisible group]]{#Dem} * [[nLab:Grothendieck's Galois theory]] * Richard Taylor, IAS, Galois representations, [pdf](http://www.math.ias.edu/~rtaylor/longicm02.pdf) *George Janelidze, Walter Tholen, extended Galois theory and dissonant morphisms