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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.

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    $\begingroup$ 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. $\endgroup$
    – Kestutis Cesnavicius
    Commented Jan 29, 2014 at 18:18
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    $\begingroup$ By $p$-extension, do you possibly mean pro-$p$-extension (or am I confused by the meaning)? $\endgroup$
    – Bobby Grizzard
    Commented Jan 29, 2014 at 18:19
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    $\begingroup$ Sorry, yes, I mean pro-p extension. $\endgroup$
    – Henri Johnston
    Commented Jan 29, 2014 at 21:22

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The answer seems to be yes. I think this was proved by Kuhlmann, Pank, and Roquette, "Immediate and purely wild extensions of valued fields", Manuscripta math. 55 (1986), 39-67. A short proof is given in Efrat's book on valuation theory, p. 203.

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