# 5 Galois Theory

The usual proof of the usual Fundamental Theorem of Galois Theory can pretty much be carried over to the general case if we can say the word “Galois”. Fortunately, the word is not too difficult to utter.

A *Galois cover* of $X$ is an element $Y \in \mathscr {FE}t_{X}$ such that $\operatorname{Aut}(Y)$ acts transitively on $F(Y)$.

Let $Y \in \mathscr {FE}t_{X}$ be Galois. Then $Y \times _ X Y$ splits as finitely many disjoint copies of $Y$, and $\operatorname{Aut}(Y)$ acts freely and transitively over the copies, where $\operatorname{Aut}(Y)$ acts as an automorphism of $\pi _1: Y \times _ X Y \to Y$ by pullback.

In Galois theory, we can construct Galois closures. We can do the same here.

Let $Y \in \mathscr {FE}t_{X}$ be connected. Then there is a Galois cover $Y'$ with a map $Y' \to Y$ such that if $Z \in \mathscr {FE}t_{X}$ is any Galois cover, then any map $Z \to Y$ factors through $Y'$.

Let $y_1, \ldots , y_ n \in F(Y)$, which gives rise to a geometric point $y = (y_1, \ldots , y_ n) \in Y^ n$. Let $Y'$ be the component of $y \in Y^ n$, mapping to $Y$ by first projection. This is an étale cover since it is a connected component of one. We claim that this works.

If $(x_1, \ldots , x_ n) \in Y'$, then $x_ i \not= x_ j$ for all $i$ and $j$.

$Y' \to X$ is Galois.

If $Z$ is Galois and $q: Z \to Y$, then it factors through $Y'$.

Fix a lift $z \in F(Z)$. Then by composing with automorphisms of $\operatorname{Aut}(Z)$, we can pick maps $q_1, \ldots , q_ n: Z \to Y$ such that $q_ i(z) = y_ i$. Then the image of $(q_1, \ldots , q_ n): Z \to Y^ n$ includes a point in $Y'$, namely $y$, and hence maps into $Y'$ by connectedness of $Z$.

□Let $(X_\alpha )_\alpha \subseteq \mathscr {FE}t_{X}$ be the poset of Galois covers of $X$, ordered under $X_\alpha \leq X_\beta$ if there is a morphism $X_\beta \to X_\alpha$. For each $X_\alpha$, pick a point $x_\alpha \in F(X_\alpha )$. Then if $X_\alpha \leq X_\beta$, we pick the morphism that sends $x_\alpha$ to $x_\beta$ for use in the pro-system. Then this pro-system pro-represents $F$.

$\mathscr {FE}t_{X}$ is naturally equivalent to the category of continuous finite $\pi _1(X, x)$-sets.

^{1}. Functions can be constructed similarly. □