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arXiv · 2602.08095

The Galois characterisation of $p$-adically closed fields -- A modern perspective

Leo Gitin·Jochen Koenigsmann·Benedikt Stock
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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, with a Galois characterisation of henselianity at the heart of the proof and without relying on 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.

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Leo Gitin, Jochen Koenigsmann, Benedikt Stock. 2026-02-08. The Galois characterisation of $p$-adically closed fields -- A modern perspective. https://arxiv.org/abs/2602.08095

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