Uchida's conjecture on open homomorphisms of Galois groups of solvably closed extensions of number fields

Claims Uchida's conjecture in full: for number fields $F_1,F_2$ and possibly infinite solvably closed Galois extensions $E_i/F_i$, every continuous open homomorphism $\mathrm{Gal}(E_1/F_1)\to\mathrm{Gal}(E_2/F_2)$ is induced by a unique field embedding $E_2\hookrightarrow E_1$, with no hypothesis on the kernel and no separate cyclotomic-compatibility assumption. The key steps derive full cyclotomic compatibility from openness, through a uniform (in the auxiliary prime) Kummer bound and Frobenius-average character tests, and then apply Hoshi's criterion. It does NOT treat function fields, finite solvable quotients (Saidi-Tamagawa), or non-open homomorphisms.

Claims Uchida's conjecture in full: for number fields $F_1,F_2$ and possibly infinite solvably closed Galois extensions $E_i/F_i$, every continuous open homomorphism $\mathrm{Gal}(E_1/F_1)\to\mathrm{Gal}(E_2/F_2)$ is induced by a unique field embedding $E_2\hookrightarrow E_1$, with no hypothesis on the kernel and no separate cyclotomic-compatibility assumption. The key steps derive full cyclotomic compatibility from openness, through a uniform (in the auxiliary prime) Kummer bound and Frobenius-average character tests, and then apply Hoshi's criterion. It does NOT treat function fields, finite solvable quotients (Saidi-Tamagawa), or non-open homomorphisms.