Given a local field is the maximal unramifield extension always finite? I was just wondering given a local field complete with non-archimedean valuation, is the maximal unramified extension always finite or could it be infinite? Any comments are appreciated!

For a finite extension of $\Bbb Q_p$ the maximal unramified extension
is infinite. It is generated by adjoining the $n$-th roots of unity
for all $n$ coprime to $p$. The Galois group is the cyclic profinite
group $\hat{\Bbb Z}$, and is naturally isomorphic to the Galois
group of the algebraic closure of $\Bbb F_p$. This is in all textbooks
on local fields, for instance Serre's.
