031. Uchida’s conjecture for open homomorphisms of Galois groups

Uchida’s conjecture for open homomorphisms of Galois groups

1 paper

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.

Papers 1

Open Homomorphisms of Global Solvably Closed Galois Groups

OpenAI · October 5, 2026

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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@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}
}

Uchida’s conjecture for open homomorphisms of Galois groups

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Papers 1

Open Homomorphisms of Global Solvably Closed Galois Groups

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