Well the question is in the title.
I asked myself this question while thinking about something in Grothendieck-Teichmüller theory. I guess class field theory gives some insight into this, or I am missing something absolutely obvious..
[Math] What can we say about center of rational absolute Galois group
absolute-galois-groupanabelian-geometrygalois-theorynt.number-theory
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For me, it's one of those questions that would not be so interesting if the answer is Yes but which would probably be very interesting if the answer is No. If not all groups are Galois groups over Q, then there is probably some structure that can be regarded as an obstruction, and then this structure would probably be essential to know about. For instance, not all groups are Galois groups over local fields -- they have to be solvable. This is by basic properties of the higher ramification filtration, which is, surprise, essential to know about if you want to understand local fields. So you could say it's an approach to finding deeper structure in the absolute Galois group. Why not just do that directly? The problem with directly looking for structure is that it's not a yes/no question, and so sometimes you lose track of what exactly you're doing (although in new and fertile subjects often you don't). So the inverse Galois problem has the advantage of being a yes/no question and the advantage that things would be really interesting if the answer is No. Unfortunately, I think the answer is expected to be Yes, though correct me if I'm wrong.
One reason we prefer the Weil group over the Galois group (at least in the local case) is that the Weil group is locally compact, thus it has "more" representations (over $\bf C$). In fact, all $\bf C$-valued characters of $Gal(\bar{\bf Q_p} / \bf Q_p)$ have finite image, where as that of $W_{\bf Q_p}$ can very well have infinite image. The same goes for general representations of these groups (recall that $\bf{GL}_n(\bf C)$ has no small subgroups.)
The global Weil group (which is much more complicated than the local one), on the other hand, is a rather mysterious object that is pretty much untouched in modern number theory as far as I can tell. Supposedly the global Langlands group used in the global Langlands correspondence should be the extension of the global Weil group by a compact group, but this is still largely conjectural.
The standard reference is Tate's "Number Theory Background" in the Corvallis volumes (available for free at ams.org). Also Brooks Roberts has notes on Weil representations available at his website.
Best Answer
The proof of triviality is a step in the famous Neukirch-Uchida theorem of anabelian geometry, which says a number field is characterized by its absolute Galois group, even functorially, in an appropriate sense. The key elementary fact is the following:
Let $k$ be a number field, $K$ an algebraic closure, and $G=Gal(K/k)$. Let $P_1$ and $P_2$ be two distinct primes of $K$ with corresponding decomposition subgroups $G(P_i)\subset G$. Then
$G(P_1)\cap G(P_2)=1.$
Once this is stated for you, it's essentially an exercise to prove.
Determining the center of $G$ becomes then completely straightforward: Suppose $g$ commutes with everything. Then for any prime $P$, $G(gP)=gG(P)g^{-1}=G(P)$. So $g$ must fix every prime, implying that it's trivial.
I think this is spelled out in the book Cohomology of Number Fields, by Neukirch, Schmidt, and Wingberg. Unfortunately, I left my copy on the plane last year.