Re: Galois group of a global field

Just some remarks:

Let k be a global field and s a separable closure of
k.

For each place v of k, let k_v be a completion k_v of
k at v, and s_v
a separable closure of k_v containing s.

Let G (resp. G_v) be the Galois group of s/k (resp.
of s_v/k_v), and
H_v the image of the restriction morphism from G_v to
G.

Isn't H_v the decomposition group of the restriction to s
of the valuation living on s_v?

Is G topologically generated by the H_v?

Let U be the closed subgroup of G generated by all H_v,
and let L be the fixed field of U. If the H_v are really the
decomposition groups, then the extension L|k is unramified
everywhere. In the case of the rationals or a rational function
field of characteristic zero one gets L=k hence U=G.

H
.