TITLE: Explicitly describable maximal unramified extension of a number field
QUESTION [17 upvotes]: This is an idle question.  Is there an example of a number field $K$ for which the maximal everywhere-unramified extension $M|K$ is of infinite degree, and yet the group $\mathrm{Gal}(M|K)$ has been completely determined ?

REPLY [5 votes]: It seems the question is not so idle after all, for the Fontaine-Mazur conjecture has something to say about it, as I discovered today in Neukirch-Schmidt-Wingberg (the NSW of Cam's answer).  To paraphrase their (10.8.13), every representation
$$
\operatorname{Gal}(M|K)\to
\operatorname{GL}_n(\mathbf{Q}_p)
$$
of the group of automorphisms of the maximal unramified extension $M$ of a number field $K$ has finite image, even if the degree $[M:K]$ is infinite.  This is certainly a very strong restriction.