Unramified extensions of quadratic number fields with Galois group $2.A_n$

TL;DR

This paper constructs infinitely many quadratic number fields with unramified Galois extensions having Galois groups isomorphic to covering groups of alternating groups, advancing the understanding of unramified extensions in number theory.

Contribution

It provides the first unramified realizations of infinitely many covering groups 2.A_n as Galois groups over quadratic fields, extending previous partial or conditional results.

Findings

Infinite families of quadratic fields with specified unramified Galois groups

First explicit constructions of 2.A_n as unramified Galois groups

Progress in the inverse Galois problem for unramified extensions

Abstract

We realize infinitely many covering groups $2.A_n$ (where $A_n$ is the alternating group) as the Galois group of everywhere unramified Galois extensions over infinitely many quadratic number fields. After several predecessor works investigating special cases or proving conditional results in this direction, these are the first unramified realizations of infinitely many of these groups.