Elementary characterisation of $\mathbb{Q}_p$ by its absolute Galois group -- A modern perspective
Leo Gitin, Jochen Koenigsmann, Benedikt Stock
TL;DR
The paper tackles the problem of characterizing fields by their absolute Galois groups, proving that a field is $p$-adically closed iff $G_K \\\cong G_{\\{\\mathbb{Q}}_p}$, using a new elementary, self-contained valuation-theoretic approach. It reconstructs a henselian valuation from $G_K$, applies a Standard Decomposition to express it as a composition of valuations, and uses Pop's Lemma together with transfer principles to force the valuation to have mixed characteristic $(0,p)$ with value group $\\mathbb{Z}$ and residue field $\\mathbb{F}_p$. The method avoids Galois cohomology and local class field theory, replacing those with explicit, elementary arguments, and it connects to recent work on perfectoid fields and model-theoretic transfer. The authors also provide a systematic, self-contained account of these methods to enable further exploration and potential extensions to higher-rank valuations in related anabelian-type questions.
Abstract
In 1927, Artin and Schreier showed that a field is real closed if and only if its absolute Galois group has order two. Inspired by this characterisation and drawing on earlier work of Neukirch, Pop conjectured the following $p$-adic analogue: a field is $p$-adically closed if and only if its absolute Galois group is isomorphic to that of $\mathbb{Q}_p$. In 1995, the conjecture was independently solved by Efrat for $p \ne 2$ and by Koenigsmann in full generality. Using novel techniques in the theory of valued fields developed over the last 25 years, we give a new elementary and self-contained proof of this theorem, which does not rely on local class field theory and Galois cohomology. We further highlight connections to the recent work of Jahnke-Kartas on perfectoid fields and model-theoretic transfer techniques. We provide a systematic account of all of our methods to encourage further investigations.