Browse › Number theory › 031 Uchida’s conjecture for open homomorphisms of Galois groups
Open Homomorphisms of Global Solvably Closed Galois Groups
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Abstract
We prove Uchida's conjecture on open homomorphisms of Galois groups. Every continuous open homomorphism between Galois groups of solvably closed Galois extensions of number fields is induced by a unique equivariant field embedding in the opposite direction. No restriction on the kernel or additional compatibility hypothesis is required.
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Related prior literature mentioned by others
- Yuichiro Hoshi, Homomorphisms of Global Solvably Closed Galois Groups Compatible with Cyclotomic Characters (Tohoku Math. J. (2) 77, 17–32, 2025; linked preprint dated September 2022). The Open Mathematics Courses exposition identifies Hoshi's equivalence between cyclotomic compatibility and existence of an equivariant field embedding as an external input to the manuscript's final reduction.
Possibly relevant literature (found by the agent, not checked)
- Alberto Corato and Mohamed Saïdi, The m-step Solvable Hom-Form of Birational Anabelian Geometry for Number Fields (arXiv:2307.07834, July 2023). Proves an m-step solvable analogue of Uchida's 1981 conditional existence theorem under hypotheses described as somewhat weaker than Uchida's, unconditional when the source field is Q; the paper's introduction discusses Uchida's local-condition result and Mao–Saïdi but does not cite this work.
- Yuichiro Hoshi, Mono-anabelian Reconstruction of Solvably Closed Galois Extensions of Number Fields (J. Math. Sci. Univ. Tokyo 29, 257–283, 2022). Its functorial reconstruction of a solvably closed extension together with its Galois action provides background for the manuscript's claim that open homomorphisms recover equivariant field embeddings.
Digests and expositions
- Open Mathematics Courses / Claude Opus 5.5, Open homomorphisms of solvably closed Galois groups (online course, undated). Eight lessons expand the manuscript's argument through logarithm ranks, Frobenius detection, Kummer bounds and residue-characteristic matching, with Uchida's and Hoshi's results named as external inputs and independent review described as incomplete. [AI-generated per source]
News and blogs
- ぶちょう, Grothendieck's 'Section Conjecture', AI claims proof of p-adic version using Shinichi Mochizuki's tools—OpenAI and the Kyoto lineage (note.com, 2026-10-08). Although mainly about family 019, the post explicitly discusses family 031's use of Hoshi's cyclotomic criterion at the endpoint of its claimed proof.
Elsewhere
- VibeMathed: Problem record
- Emergent Mind: family 031
Independent sites; their content is not reviewed here.
Cite
@misc{OAI:Open-Homomorphisms-of-Global-Solvably-Closed-Galois-Groups-October-5-2026,
author = {{OpenAI}},
title = {{Open Homomorphisms of Global Solvably Closed Galois Groups}},
howpublished = {OpenAI Math Release preprint
\href{https://github.com/openai/math/blob/main/preprints/Open-Homomorphisms-of-Global-Solvably-Closed-Galois-Groups-October-5-2026/open-homomorphisms-solvably-closed-galois-groups.pdf}{OAI:Open-Homomorphisms-of-Global-Solvably-Closed-Galois-Groups-October-5-2026}},
year = {2026}
}Family 031: Uchida’s conjecture for open homomorphisms of Galois groups
Proves Uchida's conjecture: every continuous open homomorphism between Galois groups of possibly infinite solvably closed Galois extensions of number fields comes from a unique equivariant field embedding in the opposite direction. No restriction on the kernel or separate cyclotomic-compatibility assumption is needed.
This is the only paper in this family.