Galois realizations with inertia groups of order two

Abstract

There are several variants of the inverse Galois problem which involve restrictions on ramification. In this paper, we give sufficient conditions that a given finite group [Formula: see text] occurs infinitely often as a Galois group over the rationals [Formula: see text] with all nontrivial inertia groups of order [Formula: see text]. Notably any such realization of [Formula: see text] can be translated up to a quadratic field over which the corresponding realization of [Formula: see text] is unramified. The sufficient conditions are imposed on a parametric polynomial with Galois group [Formula: see text] — if such a polynomial is available — and the infinitely many realizations come from infinitely many specializations of the parameter in the polynomial. This will be applied to the three finite simple groups [Formula: see text], [Formula: see text] and [Formula: see text]. Finally, the applications to [Formula: see text] and [Formula: see text] are used to prove the existence of infinitely many optimally intersective realizations of these groups over the rational numbers (proved for [Formula: see text] by the first author in [J. König, On intersective polynomials with nonsolvable Galois group, Comm. Alg. 46(6) (2018) 2405–2416.