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.

Manuscript

Open Homomorphisms of Global Solvably Closed Galois Groups

October 5, 2026 · source · cite as \cite{OAI:Open-Homomorphisms-of-Global-Solvably-Closed-Galois-Groups-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.

Result 031 · Number theory. Imported from openai/math, a collection of manuscripts produced by an OpenAI model; results are at different stages of verification. Discussion below is by Lemma members.