The m-step Solvable Mono-anabelian Geometry of Number Fields

TL;DR

This paper develops a group-theoretic algorithm to reconstruct number fields and their maximal m-step solvable extensions from specific quotients of their absolute Galois groups, advancing anabelian geometry methods.

Contribution

It introduces a novel algorithm for reconstructing number fields from their Galois group quotients, extending previous work to m-step solvable extensions.

Findings

Reconstruction algorithm for general number fields from m+9-step quotients.

Specialized algorithm for imaginary quadratic fields and Q from 6-step quotients.

Proof of correctness for the proposed reconstruction methods.

Abstract

The goal of this paper is to develop a group-theoretic algorithm, to reconstruct a number field (together with its maximal m-step solvable ex- tension for some positive integer m \geq 3) from the maximal m+9-step solv- able quotient of its absolute Galois group. If K is an imaginary quadratic field or Q, we establish a group-theoretic reconstruction algorithm of K from the maximal 6-step solvable quotient of its absolute Galois group.