The Galois characterisation of $p$-adically closed fields -- A modern perspective
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arXiv
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| Format: | Preprint |
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2026
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| author | Gitin, Leo Koenigsmann, Jochen Stock, Benedikt |
| author_facet | Gitin, Leo Koenigsmann, Jochen Stock, Benedikt |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_08095 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | The Galois characterisation of $p$-adically closed fields -- A modern perspective Gitin, Leo Koenigsmann, Jochen Stock, Benedikt Number Theory Commutative Algebra Logic 11S20, 12L12 (Primary) 11U09, 12L10 (Secondary) 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. |
| title | The Galois characterisation of $p$-adically closed fields -- A modern perspective |
| topic | Number Theory Commutative Algebra Logic 11S20, 12L12 (Primary) 11U09, 12L10 (Secondary) |
| url | https://arxiv.org/abs/2602.08095 |