# Number Theory Lecture 8

The lecture NumTh8

### Excerpts

Profinite groups: inverse limit of fintie groups.  Some common examples

1. profinite completion $\hat G := \varprojlim_i G/H_i$, $H_i$ all normal subgroups of finite index.
2. pro-p completion $\hat G_p := \varprojlim_i G/F_i$, $F_i$ all normal subgroups of index a power of $p$.

Another example.  For $E/F$ an algebraic extension.  Look at all finite Galois extensions $K_i/F \subset E/F$.  These for a directed system under $K_1 \le K_2$ if $K_1 \subset K_2$.  Now we said earlier if $F \subset K_1 \subset K_2$ are all Galois extensions, then elements of $Gal(K_2/F)$ restrict to $Gal(K_1/F)$.  So we get an inverse system.  And infact

$Gal(E/F) \cong \varprojlim_i Gal(K_i/F)$

Also there is the fundamental theorem of Galois theory that says closed subgroups of a Galois group $G = Gal(E/F)$ are in bijective correspondence with intermediate fields: $E \supset K \mapsto Gal(E/K) \subset G$.

The rest of the lecture gives an example of how this formalism occurs in previous constructions.  In particular another view of the contruction of the Frobenius element.