Mono-anabelian Reconstruction of Solvably Closed Galois Extensions of Number Fields

Abstract

A theorem of Uchida asserts that every continuous isomorphism between the Galois groups of solvably closed Galois extensions of number fields arises from a unique isomorphism between the solvably closed Galois extensions. In particular, the isomorphism class of a solvably closed Galois extension of a number field is completely determined by the isomorphism class of the associated Galois group. On the other hand, neither the statement of this theorem nor the proof of this theorem yields an "explicit reconstruction" of the given solvably closed Galois extension. In the present paper, we establish a functorial "grouptheoretic" algorithm for reconstructing, from the Galois group of a solvably closed Galois extension of a number field, the given solvably closed Galois extension equipped with the natural Galois action.