Let $K$ be a complete, discretely valued field with (let's say) perfect residue field $k$. We have a unique maximal unramified extension $K^{unr}$ of $K$ and a unique maximal tamely ramified extension $K^{tame}$ of $K$ and hence short exact sequences
$1 \rightarrow Gal(K^{sep}/K^{unr}) \rightarrow Gal(K^{sep}/K) \rightarrow Gal(K^{unr}/K) \rightarrow 1$
and
$1 \rightarrow Gal(K^{tame}/K^{unr}) \rightarrow Gal(K^{tame}/K) \rightarrow Gal(K^{unr}/K) \rightarrow 1$.
In the second case, the normal subgroup is abelian and I know exactly what the action of the quotient on it is: the tame cyclotomic character. Therefore if it splits, I know its structure as an explicit semidirect product.
In the most famous case, $k$ is finite, so $Gal(K^{unr}/K) = Gal(k^{sep}/k) \cong \widehat{\mathbb{Z}}$ is a projective profinite group, so both sequences certainly split. This means that I (and lots of other people) do know the structure of the tame Galois group explicitly: it is $\prod_{\ell \neq p} \mathbb{Z}_{\ell}(1) \rtimes \widehat{\mathbb{Z}}$. Similarly the first sequence splits so there is a totally ramified extension $L/K$ such that $K^{sep}/L$ is unramified. Moreover, this is a very useful fact: it follows for instance that any abelian variety over $K$ with potentially good reduction acquires good reduction over a totally ramified base extension.
What is known in general? We have $Gal(K^{unr}/K) = Gal(k^{sep}/k)$, but if $k$ is almost anything else reasonable — e.g. a local or global field, or the function field of a variety — then its absolute Galois group certainly will not be projective. What is known about the splitting of these two short exact sequences in general, and especially about the class $\eta \in H^2(Gal(K^{unr}/K),Gal(K^{tame}/K^{unr}))$ defined by the second sequence? Is there information on how the analogues of the above results do / do not work out if these sequences do not split?
Best Answer
The second sequence always split. This is proved in a paper by Kuhlmann, Pank, and Roquette, manuscripta mathematica 55 (1986), 39-77.