Uchida’s conjecture for open homomorphisms of Galois groups

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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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Theorem 1.1

In Open Homomorphisms of Global Solvably Closed Galois Groups, 5 October 2026

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Let F_1,F_2 be number fields, and let E_i/F_i be possibly infinite solvably closed Galois extensions. Every continuous open homomorphism

\alpha:\Gal(E_1/F_1)\longrightarrow\Gal(E_2/F_2)

is induced by a unique field embedding j:E_2\hookrightarrow E_1.

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Open Homomorphisms of Global Solvably Closed Galois Groups

5 October 2026, current edition.

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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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