TITLE: Absolute Galois group of $\Bbb Q_p$ while varying $p$
QUESTION [9 upvotes]: $\newcommand{\Q}{\Bbb Q}
\newcommand{\N}{\Bbb N}
\newcommand{\R}{\Bbb R}
\newcommand{\Z}{\Bbb Z}
\newcommand{\C}{\Bbb C}
\newcommand{\A}{\Bbb A}
\newcommand{\ab}{\mathrm{ab}}
\newcommand{\Gal}{\mathrm{Gal}}
\newcommand{\prolim}{\varprojlim}
$
Let $p, q$ be two distinct prime numbers.

Are the groups $\Gal(\overline{\Q_p} / \Q_p)$ and $\Gal(\overline{\Q_q} / \Q_q)$ isomorphic (as abstract groups)? Are they isomorphic as topological groups?

Thoughts:
1) The absolute Galois group of a number field $K / \Q$ determines the number field, by Neukirch–Uchida theorem. But this does not apply to local fields.
2) This reference explains that the absolute Galois group of a $p$-adic number field (with $p$ fixed) only determines the field up to elementarily equivalence. But here I'm asking for various $p$.
3) I think these two profinite groups are homeomorphic, by Brouwer theorem (every non-empty second countable totally disconnected compact Hausdorff topological space without isolated points is homeomorphic to the Cantor middle-third set). But I'm asking about (topological) group isomorphisms.
Thank you!

REPLY [4 votes]: Cohomological methods (such as in the book of Neukirch-Schmidt-Wingberg) give the most direct proof that the two groups cannot be isomorphic as profinite groups. For convenience, let us call $K$ a local $p$-adic field if $K$ is a finite extension of $\mathbf Q_p$, say of degree $N$, and denote its absolute Galois group by $G_K$. Let $\delta =1$ if $K$ contains a primitive $p$-th root of unity, $0$ otherwise. Then for any prime $l, dim_{\mathbf F_l} H^1(G_K,\mathbf Z/l\mathbf Z)=1+N+\delta$ if $l=p,1+\delta$ otherwise, which already shows what we want. 
Another reason (also of a cohomological nature) lies in the structure of the maximal pro-$p$-quotient of $G_K$, which is pro-$p$-free on $N+1$ generators if $\delta=0$ (Chafarevitch), minimally generated by $N+2$ generators and $1$ explicit relation if $\delta=1$ (Demushkin for $p$ odd, Labute for $p=2$). 
Actually, an almost complete description of $G_K$ by generators and relations is known : $G_K$ can be generated by $N+3$ generators $\sigma, \tau, x_0 ,..., x_N$ satisfying a "tame" relation $\sigma \tau \sigma^{-1}= \tau^q$, where $q$ is the cardinal of the residual field of $K$, as well as an explicit "wild" relation whose expression depends on the parity of $N$ (Jannsen-Wingberg for $p$ odd, Diekert for $p=2$ and $K(\zeta_4)/K$ unramified). Curiously, the remaining case is still unknown. All these results , and many more, can be found in chapter VII, §5, op. cit.