Infinite Galois Theory - Local Fields

15 Infinite Galois Theory

Definition 15.1 (Infinite Galois definitions).

$L ∕ K$ is separable if $\forall α \in L$ , the minimal polynomial $f_{α} (X) \in K [X]$ for $α$ is separable.
$L ∕ K$ is normal if $f_{α} (X)$ splits in $L$ for all $α \in L$ .
$L ∕ K$ is Galois if it is separable and normal. Write $Gal (L ∕ K) : = {Aut}_{K} (L)$ in this case. If $L ∕ K$ is a finite Galois extension, then we have a Galois correspondence:
$\begin{array}{l} {subextensions K \subseteq K^{'} \subseteq L} & \leftrightarrow {subgroups of Gal (L ∕ K)} \\ K^{'} & \mapsto Gal (K ∕ K^{'}) \end{array}$

Let $(I, \leq)$ be a poset. Say $I$ is a directed set if for all $i, j \in I$ , there exists $k \in I$ such that $i \leq k$ , $j \leq k$ .

Example.

Any total order (for example $(ℕ, \leq)$ ).
$ℕ_{\geq 1}$ ordered by divisibility.

Definition 15.2. Let $(I, \leq)$ be a directed set and ${(G_{i})}_{i \in I}$ a collection of groups together with maps $φ_{i j} : G_{j} \to G_{i}$ , $i \leq j$ such that:

$φ_{i k} = φ_{i j} \circ φ_{j k}$ for any $i \leq j \leq k$
$φ_{i i} = id$

Say $({(G_{i})}_{i = 1}, φ_{i j})$ is an inverse system. The inverse limit of $(G_{i}, φ_{i})$ is

\lim_{\overset{[}{i}] \leftarrow} G_{i} = {{(g_{i})}_{i \in I} \in \prod_{i \in I} G_{i} | φ_{i j} (g_{j}) = g_{i}} .

Remark.

$(ℕ, \leq)$ recovers the previous set.
There exist projection maps $φ_{j} : \lim_{\overset{[}{i} \in I] \leftarrow} G_{i} \to G_{j}$ .
$\lim_{\overset{[}{i} \in I] \leftarrow} G_{i}$ satisfies a universal property.
Assume $G_{i}$ finite. Then the profinite topology on $\lim_{\overset{[}{i} \in I] \leftarrow} G_{i}$ is the weakest topology such that $φ_{j}$ are continuous for all $j \in I$ .

Proposition 15.3. Assuming that:

$L ∕ K$ Galois

Then

(i) The set $I = {F ∕ K finite | F \subseteq L, F Galois}$ is a directed set under $\subseteq$ .
(ii) For $F, F^{'} \in I$ , $F \subseteq F^{'}$ there is a restriction map ${res}_{F, F^{'}} : Gal (F^{'} ∕ K) ↠ Gal (F ∕ K)$ and the natural map
$Gal (L ∕ K) \to \lim_{\overset{[}{F} \in I] \leftarrow} Gal (F ∕ K)$

is an isomorphism.

Proof. Example Sheet 4. □