Unramified extensions of quadratic number fields with Galois group $S{L}_2(7)$

Abstract

We provide an infinite family of quadratic number fields with everywhere unramified Galois extensions of Galois group $S{L}_2(7)$. To my knowledge, this is the first instance of infinitely many such realizations for a perfect group which is not generated by involutions, a property which makes it difficult to approach for the problem in question and leads to somewhat delicate local-global problems in inverse Galois theory.

Adapted from the published version ("Unramified extensions of quadratic number fields with certain perfect Galois groups", Int. J. Number Theory 19(3) (2023), 639-653) in which the proof of the additionally treated case $G=2.A_7$ was irreparably flawed, and has therefore been removed here