In Chapter II/§7 of Neukirch "Algebraic Number Theory" an unramified extension of a field $K$ which is Henselian with respect to some (not necessarily discrete) non-archimedean valuation $v$ is defined as an extension $L/K$ such that the extension $l/k$ of residue fields is separable and $[L:K]=[l:k]$.

As Neukirch points out, this definition still makes sense if $v$ only has a unique extension to $L$ instead of $K$ being Henselian. So my question is: Do the results of §7 (i.e. composites of unramified extensions are still unramified, the maximal residue field of the maximal unramified intermediate extension is the separable closure $l^{sep}$ of $k$ in $l$)

  1. make sense (e.g. is $v$ uniquely entensible to $LL'$ if it is to $L$ and $L'$, otherwise Neukirch doesn't even define whether $LL'/K$ is unramified)
  2. still hold

under this weaker assumption? And what about their analogues in the tamely ramified case?

Neukirch makes extensive use of Hensel's Lemma, which of course isn't fully available in the non-Henselian case. However, if I'm correct, Hensel's Lemma should still hold for polynomials $f$ such that $v$ extends uniquely to the splitting field of $f$. So I think some of the arguments can be carried over to the non-Henselian case.

  • 2
    $\begingroup$ Henselian fields are exactly those for which Hensel's lemma holds. $\endgroup$ – Adam Hughes May 26 '17 at 19:31

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