A note on open homomorphisms between global solvably closed Galois groups

Abstract

In the present paper, we study continuous open homomorphisms between the Galois groups of solvably closed Galois field extensions of number fields. In particular, we discuss Uchida's conjecture that asserts that an arbitrary continuous open homomorphism between the Galois groups of solvably closed Galois field extensions of number fields arises from a homomorphism between the given Galois field extensions. In the present paper, we prove that this conjecture is equivalent to the assertion that if the Galois group of a Galois field extension of a number field is isomorphic to an open subgroup of the maximal prosolvable quotient of the absolute Galois group of the field of rational numbers, then, for all prime numbers l and all but finitely many prime numbers p, the given Galois extension field contains l roots of the polynomial t^l - p. Moreover, we prove that this conjecture is also equivalent to the assertion that if the Galois group of a Galois field extension of an absolutely Galois number field is isomorphic to an open subgroup of the maximal prosolvable quotient of the absolute Galois group of the field of rational numbers, then the given Galois extension field is absolutely Galois.