---
title: "Uchida’s conjecture for open homomorphisms of Galois groups"
authors:
  - "OpenAI (internal model)"
year: 2026
venue: "openai/math manuscript collection (GitHub), result family 031; unreviewed"
arxiv: null
doi: null
source: "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"
topics:
  - "openai-math"
  - "ai-generated-math"
  - "number-theory"
  - "unformalized"
seed_rank: 1891
seed_batch: "openai-math-2026-10-07"
reviewed: "2026-10-07"
pool: "maths-foundations"
relevance_score: 7
lineage: ai-mathematical-reasoning
cites:
  - title: "Open Homomorphisms of Global Solvably Closed Galois Groups"
    url: "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"
    year: 2026
    arxiv: null
    doi: null
---

# Uchida’s conjecture for open homomorphisms of Galois groups

## One-sentence takeaway

OpenAI's result family 031 (Number theory) claims: 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.

## Why it matters here

Pure mathematics with no direct engine use; it is in the library as part of the openai/math corpus, which records what an AI research model claims to have proved. More broadly, every card in this batch is evidence for the `ai-mathematical-reasoning` thread: per the repo README, an unreleased internal OpenAI model produced it, using about three hours of ChatGPT Pro thinking compute per result on average, across roughly 4,000 posed problems.

## Key ideas

- Family summary (repo): No restriction on the kernel or separate cyclotomic-compatibility assumption is needed.
- *Open Homomorphisms of Global Solvably Closed Galois Groups*: 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.

## Caveats

- Unreviewed: these are AI-generated manuscripts posted to GitHub with no peer review or arXiv deposit. The repo README says results are "at different stages of verification" and that "some of the unformalized results could have issues."
- This card restates the repo's family summary, manuscript abstracts and Lean scope notes. It is not an independent check of the proofs, and the scope of each theorem is exactly as stated there.
- Lean: not formalized. The repo has no Lean development for this family, so the claim rests on the prose manuscript(s) alone.

## Links

- Manuscript: [Open Homomorphisms of Global Solvably Closed Galois Groups](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)
- Family entry: https://github.com/openai/math/blob/main/CONTENTS.md
- Overview PDF: https://github.com/openai/math/blob/main/overview.pdf
- Repo: https://github.com/openai/math