Result 031, Number theory

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.

Proof

The bigger picture

Why it matters

Can maps between groups of arithmetic symmetries come only from maps between the underlying fields? The manuscript reports that they do for a broad class of extensions of number fields, with uniqueness as well.

What changes?

The claim covers possibly infinite solvably closed Galois extensions of number fields, which are finite extensions of the rational numbers. Solvably closed means having no nontrivial finite Galois extension with a solvable symmetry group. A Galois group consists of field symmetries fixing the base field. Every continuous open homomorphism, a group map respecting the topology and sending open sets to open sets, reportedly comes from a unique field embedding in the reverse direction that respects these symmetry actions.

What does that help mathematicians do?

This would let researchers translate such group maps into uniquely determined field embeddings, ruling out maps satisfying these hypotheses that have no underlying field interpretation. No restriction is imposed on the kernel, the symmetries the map collapses. Nor must compatibility with the action on roots of unity be assumed separately. Thus the claimed correspondence applies beyond injective maps without requiring that extra arithmetic information as input.

Are there practical applications?

The immediate value is foundational: the result links a description by symmetry groups to a description by fields and their embeddings. For researchers studying how much arithmetic these groups retain, it would provide a precise reconstruction principle for maps, not just a comparison of individual groups. The supplied sources describe no practical deployment.

This section was generated by GPT-6 Astra Medium. This explanation is based on the result summary and manuscript abstracts below. This context is separate from OpenAI's source text.

Manuscript

Open Homomorphisms of Global Solvably Closed Galois Groups

October 5, 2026 23 pages

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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Cite (BibTeX) 
@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}
}

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An Emergent Mind Labs project. Emergent Mind is not affiliated with OpenAI. None of these results has been peer reviewed. Cite the manuscripts themselves, using the BibTeX on each result's page.