On unramified solvable extensions of small number fields

TL;DR

This paper studies unramified solvable extensions of small number fields, focusing on minimal degrees of Galois extensions with prescribed solvable Galois groups, and improves bounds for certain classes like nilpotent groups.

Contribution

It provides improved bounds on the minimal degree of Galois number fields with unramified solvable extensions, especially for nilpotent groups.

Findings

Enhanced bounds for degrees of number fields with unramified solvable extensions

Specific results for nilpotent Galois groups

Advances in understanding Galois extensions with prescribed solvable groups

Abstract

We investigate unramified extensions of number fields with prescribed solvable Galois group and certain extra conditions. In particular, we are interested in the minimal degree of a number field $K$, Galois over $\mathbb{Q}$, such that $K$ possesses an unramified $G$-extension. We improve the best known bounds for the degree of such number fields $K$ for certain classes of solvable groups, in particular nilpotent groups.