This relates to this question:
[Existence of maximal totally ramified extensions of an arbitrary CDVF](https://mathoverflow.net/questions/17846/existence-of-maximal-totally-ramified-extensions-of-an-arbitrary-cdvf)


Let $K$ be a local field with finite residue field of characteristic $p>0$. Does there exist a maximal totally ramified $p$-extension of $K$? In other words, if $K^{tame}$ is the maximal tamely ramified extension of $K$, does the short exact sequence
$$
1 \longrightarrow Gal(K^{sep}/K^{tame}) \longrightarrow Gal(K^{sep}/K) \longrightarrow Gal(K^{tame}/K) \longrightarrow 1
$$
split? My guess would be that it doesn't, but maybe there is some trick I am missing here.



Existence of maximal totally ramified $p$-extension of a local field
The answer seems to be yes. I think this was proved by Kuhlmann, Pank, and Roquette, ["Immediate and purely wild extensions of valued fields"](http://link.springer.com/article/10.1007%2FBF01168612), Manuscripta math. 55 (1986), 39-67. A short proof is given in Efrat's book on valuation theory, p. 203.


