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

TL;DR

This paper provides a new, elementary proof of Pop's conjecture characterizing p-adically closed fields via their Galois groups, avoiding Galois cohomology and emphasizing valued field techniques.

Contribution

It offers a novel, self-contained proof of the Galois characterization of p-adically closed fields, connecting to recent developments in perfectoid fields and model theory.

Findings

Proved Pop's conjecture with an elementary approach.

Connected Galois characterizations to valued field theory.

Highlighted links to perfectoid fields and model theory.

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.