Uchida's conjecture on open homomorphisms of Galois groups
From OpenAI Aftermath
Uchida's conjecture on open homomorphisms of Galois groups asserts that every continuous homomorphism with open image between the Galois groups of solvably closed Galois extensions of number fields comes from a field embedding in the opposite direction. It extends the Neukirch–Uchida theorem, which recovers number fields from isomorphisms of their Galois groups, from isomorphisms to open homomorphisms. A manuscript dated 5 October 2026 in OpenAI's public mathematics collection (github.com/openai/math) claims a proof. This page summarizes the published claim and lists publicly available follow-up work. Statements are the respective authors' claims; the publication of a paper is not presented here as independent verification.
Background
- As recounted in the introduction of the manuscript:
- An algebraic extension of a number field is solvably closed if it has no nontrivial abelian extension; algebraic closures and maximal prosolvable extensions are examples. Neukirch's work on normal number fields began the reconstruction of number fields from their Galois groups; Uchida, Iwasawa and Ikeda extended it to arbitrary number fields, and Uchida proved the isomorphism theorem for arbitrary solvably closed Galois extensions of number fields.
- Uchida formulated the conjecture for open homomorphisms in 1981. In the same paper he proved it when the source number field is the field of rational numbers, proved uniqueness in general, and proved existence under a local condition on decomposition groups.
- Hoshi showed that an open homomorphism is induced by a field embedding if and only if it preserves the full cyclotomic character, and later reformulated the conjecture in terms of radicals and the normality of the field fixed by a kernel. Saïdi and Tamagawa, and Mao and Saïdi, treated related questions for finite-step solvable quotients.
The original result
- Claim. Theorem 1.1 of Open Homomorphisms of Global Solvably Closed Galois Groups states that, for number fields F1 and F2 and possibly infinite solvably closed Galois extensions Ei of Fi, every continuous open homomorphism from Gal(E1/F1) to Gal(E2/F2) is induced by a unique field embedding of E2 into E1. The kernel is unrestricted, and no separate cyclotomic-compatibility hypothesis is assumed.
- Method, as described: the proof uses Hoshi's cyclotomic criterion at its endpoint. Its two main inputs are a statement that an additive ℓ-adic character of the absolute Galois group of a number field is determined by its values on any infinite set of primes of degree one, proved through a special case of the ℓ-adic Waldschmidt–Masser theorem with Laurent's interpolation determinants, and a refinement of Uchida's Kummer argument that bounds the number of missed target primes independently of the auxiliary prime ℓ. The map is reduced to open homomorphisms onto the Galois group of the maximal prosolvable extension of the rationals, shown to be surjective and compatible with cyclotomic logarithms, and finally identified with restriction up to conjugation.
- Source (openai/math repository, commit
7ef5c0c): Open Homomorphisms of Global Solvably Closed Galois Groups, dated 5 October 2026. - Formalization. The repository has no Lean scope note for this family.
Public follow-up work
| Date | Work | Relation | Public claim | Public status (checked 29 October 2026; seminar row 31 October 2026) |
|---|---|---|---|---|
| 2026-10-08 | ぶちょう, a post on note.com about OpenAI's claims on Grothendieck's section conjecture and their Kyoto sources, in Japanese | Remark (blog post) | Mainly about the collection's p-adic section conjecture; it also notes that this manuscript uses Yuichiro Hoshi's cyclotomic criterion at the last step of its claimed proof, quoting the sentence "Our proof uses the cyclotomic criterion at its endpoint". | Blog post of 8 October 2026. |
| 2026-10-11 | AI Proved What?, episode 031 Groups remember fields?, remade version of 25 October 2026 | Reading (exposition) | Video explainer of the claim that every continuous homomorphism with open image between Galois groups of solvably closed Galois extensions of number fields is induced by a unique field embedding, with no condition on its kernel, and of the route through Hoshi's cyclotomic criterion. The episode states that the result concerns number fields and solvably closed extensions only, that the analogue for p-adic fields is false even for isomorphisms, and that homomorphisms without open image are not covered. | Web page and video; the site states that its scripts and animations are produced with AI assistance and that the narration is an AI-generated voice. Status record last checked 27 October 2026: posted as a preprint: OpenAI, dated October 5, 2026, released October 6, 23 pages; under expert review: not yet; peer reviewed: not yet; checked in Lean: no. The page states that it found no public response to the claim. |
| 2026-10-15 | Open homomorphisms of solvably closed Galois groups, a course on Open Mathematics Courses (kokunoyumeto/open-math-courses-public) | Reading (exposition) | Eight lessons: solvably closed fields and Uchida's conjecture, counting irreducible components, a logarithm rank lemma, additive characters and Frobenius detection, normalizer extension and Frobenius generation, Kummer classes and S-units, matching residue characteristics, and identifying the homomorphism. The course uses two theorems of Uchida and Hoshi's criterion without proof. | The course states that it was written and self-checked by Claude Opus 5.5 (Anthropic) and that independent review is not complete. |
| 2026-10-16 | J. Jardine, Lattice24, referee review rank_394 in JJardine919/openai-math-audit (archived at Zenodo 10.5281/zenodo.23422995) | Review (AI-assisted audit) | Reports no gap. The reviewer read all sections and checked the internal argument step by step, including the logarithm rank lemma and the Frobenius detection. Its one minor note: the two anabelian inputs, Uchida's 1981 local correspondence and Hoshi's 2025 criterion, are used as black boxes that it did not check against their sources, and the manuscript only sketches why the version of Uchida's result it states follows from his case analysis. | Part of an audit of the collection's manuscripts without Lean that the audit describes as complete. The reviews are published under CC BY 4.0. |
| 2026-10-25 | Y. Hoshi and S. Tsujimura, RIMS Kyoto, talk On open homomorphisms of global solvably closed Galois groups announced in Taming the Flow: Critical reports on AI-generated mathematics, an online seminar of the Arithmetic and Homotopic Galois Theory research network organized by B. Collas, RIMS Kyoto; announcement page last updated 25 October 2026 | Reading, seminar talk announced | The abstract recalls the Neukirch–Uchida theorem, Uchida's result that the maximal prosolvable quotients already suffice, and Uchida's conjecture that every open homomorphism between Galois groups of solvably closed extensions of global fields is induced by a field embedding. The talk's references include Hoshi's 2025 paper on homomorphisms compatible with cyclotomic characters, and the page links a copy of the manuscript of 5 October 2026 as an AI-generated target note. | Announced; date to be announced. Read on 31 October 2026. |
Notes
- The background summarizes the manuscript's own account of earlier work.
- The list covers follow-ups found in public sources up to the date above; it is not a priority or completeness claim.
