Unramified extensions of quadratic number fields with certain perfect Galois groups

Abstract

We provide infinite families of quadratic number fields with everywhere unramified Galois extensions of Galois group SL 2 (7) and 2.A 7 , respectively. To my knowledge, these are the first instances of infinitely many such realizations for perfect groups which are not generated by involutions, a property which makes them difficult to approach for the problem in question and leads to somewhat delicate local-global problems in inverse Galois theory.1 Or more generally, by involutions, although allowing this does not seem to make the problem much easier.