Quadratic number fields with unramified $SL_2(5)$-extensions

TL;DR

This paper constructs the first known infinite families of quadratic number fields with unramified Galois extensions having Galois group $SL_2(5)$, advancing understanding of unramified extensions in number theory.

Contribution

It provides the first infinite families of quadratic fields with unramified $SL_2(5)$-extensions and real-quadratic fields with specific unramified Galois extensions, improving previous results.

Findings

First infinite family of quadratic fields with unramified $SL_2(5)$-extensions

First infinite family of real-quadratic fields with unramified perfect Galois extensions not generated by involutions

New existence results for quintic fields with squarefree discriminant and local conditions

Abstract

Continuing the line of thought of an earlier work, we provide the first infinite family of quadratic number fields with everywhere unramified Galois extensions of Galois group $SL_2(5)$, the (unique) smallest nonsolvable group for which this problem was previously open. Our approach also improves upon previous work by yielding the first infinite family of real-quadratic fields possessing an unramified Galois extension whose Galois group is perfect and not generated by involutions. Our result also amounts to a new existence result on quintic number fields with squarefree discriminant and additional local conditions.