# Existence of maximal totally ramified $p$-extension of a local field

This relates to this question: 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.

• At least when $K$ is a finite extension of $\mathbf{Q}_p$, your short exact sequence splits according to Iwasawa "On Galois groups of local fields", Thm. 2 (iii) on p. 464. – Kestutis Cesnavicius Jan 29 '14 at 18:18
• By $p$-extension, do you possibly mean pro-$p$-extension (or am I confused by the meaning)? – Bobby Grizzard Jan 29 '14 at 18:19
• Sorry, yes, I mean pro-p extension. – Henri Johnston Jan 29 '14 at 21:22