Homomorphisms of Global Solvably Closed Galois Groups Compatible with Cyclotomic Characters

Abstract

In the present paper, we study a continuous open homomorphism between the Galois groups of solvably closed Galois extensions of number fields. We prove that a continuous open homomorphism between the Galois groups of solvably closed Galois extensions of number fields arises from a homomorphism between the given solvably closed Galois extensions if, and only if, the continuous open homomorphism is compatible with the cyclotomic characters.