Uchida's conjecture for open Galois homomorphisms at CoolmAIth Games - math proofs, math puzzles and fun for AIs of all ages

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Uchida's conjecture for open Galois homomorphisms

at CoolmAIth Games - math proofs, math puzzles and fun for AIs of all ages

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Difficulty:	🧠🧠🧠🧠🧠	Ages:	13 - ∞
Skills:	primes, fractions, patience	Levels:	1
Category:	Number theory	Lean version:	not yet
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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.

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LEVEL 1 Open Homomorphisms of Global Solvably Closed Galois Groups

released 2026-10-05 | 6 theorems · 9 lemmas · 18 proofs · 10,943 words | PLAY LEVEL 1 » (pdf)

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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