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Uchida's conjecture on open homomorphisms of Galois groups of solvably closed extensions of number fields

Let Ei/FiE_i/F_i (i=1,2i=1,2) be Galois extensions of number fields with each EiE_i solvably closed (no nontrivial abelian extension), for example algebraic closures or maximal prosolvable extensions, and write Gi=Gal(Ei/Fi)G_i=\mathrm{Gal}(E_i/F_i). The Neukirch-Uchida theorem says that isomorphisms G1≅G2G_1\cong G_2 come from field isomorphisms. Uchida (1981) proved the analogous statement for open homomorphisms when the source field is Q\mathbb Q, proved uniqueness in general, and proved existence under a local condition on decomposition groups, and conjectured the general case. Is every continuous homomorphism α:G1→G2\alpha:G_1\to G_2 with open image induced by a field embedding j:E2↪E1j:E_2\hookrightarrow E_1, i.e. g∘j=j∘α(g)g\circ j=j\circ\alpha(g) for all g∈G1g\in G_1?

Result
Proved(see note)
Status
Candidate (review pending)
AI contribution
AI-discovered
Method
Argument
Field
Anabelian geometry; Galois groups of number fields
Posed by
Koji Uchida, Homomorphisms of Galois groups of solvably closed Galois extensions, J. Math. Soc. Japan 33 (1981), p. 595
Year posed
1981
Years open
45y
Solved
2026-10-05
Model
Unreleased internal OpenAI model
Vendor
OpenAI
Collaborators
—
Verification
Unreviewed
Publication
Announced
Collection
OpenAI math release (October 2026), version adc7f12
Significance
22 / 100
Disclosed cost
—
Wikipedia
No dedicated article

What was actually shown

Claims Uchida's conjecture in full: for number fields F1,F2F_1,F_2 and possibly infinite solvably closed Galois extensions Ei/FiE_i/F_i, every continuous open homomorphism Gal(E1/F1)→Gal(E2/F2)\mathrm{Gal}(E_1/F_1)\to\mathrm{Gal}(E_2/F_2) is induced by a unique field embedding E2↪E1E_2\hookrightarrow E_1, with no hypothesis on the kernel and no separate cyclotomic-compatibility assumption. The key steps derive full cyclotomic compatibility from openness, through a uniform (in the auxiliary prime) Kummer bound and Frobenius-average character tests, and then apply Hoshi's criterion. It does NOT treat function fields, finite solvable quotients (Saidi-Tamagawa), or non-open homomorphisms.

What the AI did

The release README says all results were produced by an unreleased internal OpenAI model, the vast majority by one fixed procedure using on average about three hours of ChatGPT Pro thinking compute per result. Its named exceptions (the Hodge conjecture for CM abelian varieties, and the Re(s) > 11/12 zero-free region, whose write-up was human edited for readability) do not concern this family. The manuscript is credited to OpenAI with no human author named. No Lean formalization of this result is in the release.

Verification

No independent mathematician has checked this yet. The main theorem was read against Uchida's conjecture as described in the manuscript (Uchida 1981, p. 595); the 1981 paper itself was not opened here. The proof ends by verifying Hoshi's cyclotomic-character criterion (Tohoku 2025), which is an input from the literature, and uses an l-adic Waldschmidt-Masser type statement for which the paper gives its own proof. No Lean formalization exists for this family.

Sources

Changelog1 change

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