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

Claims Uchida's conjecture in full: for number fields $F_1,F_2$ and possibly infinite solvably closed Galois extensions $E_i/F_i$ , every continuous open homomorphism $\mathrm{Gal}(E_1/F_1)\to\mathrm{Gal}(E_2/F_2)$ is induced by a unique field embedding $E_2\hookrightarrow E_1$ , 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.

Exact FrontierDelta

Prior state unknown→proved

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Occurred: Oct 5, 2026

Delta type: SOURCE CLAIM

Assumptions: VibeMathed verification: unreviewed. Publication: announcement. AI contribution: ai-discovered. VibeMathed editorial classifications, scores, notes, relations, and dataset structure are CC BY 4.0. Source statements and linked content retain their own rights.

Canonical aliases: Uchida's conjecture on open homomorphisms of Galois groups of solvably closed extensions of number fields · Uchida's open homomorphism conjecture

Confidence: Not scored

Registry verification: unreviewed · announcement · candidate

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

specialist registry · passed

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.

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Uchida, Homomorphisms of Galois groups of solvably closed Galois extensions (1981)

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

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Primary source named by the VibeMathed record. Presence in this import is not an independent mathematical verification.

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VibeMathed record: Uchida's conjecture on open homomorphisms of Galois groups of solvably closed extensions of number fields evidence for this event

Uchida, Homomorphisms of Galois groups of solvably closed Galois extensions (1981) evidence for this event

Uchida's conjecture on open homomorphisms of Galois groups of solvably closed extensions of number fields evidence for this event

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