# Galois realizations with inertia groups of order two

## TL;DR

This paper establishes conditions under which certain finite groups can be realized infinitely often as Galois groups over the rationals with all nontrivial inertia groups of order two, using parametric polynomials and specialization techniques.

## Contribution

It provides new sufficient conditions for infinite Galois realizations with inertia groups of order two, especially for groups like A_5, PSL_2(7), and PSL_3(3).

## Key findings

- Infinite realizations of G with inertia groups of order two are possible under certain polynomial conditions.
- Applications to specific groups demonstrate the existence of infinitely many such Galois extensions.
- Results include the existence of infinitely many optimally intersective realizations for certain groups.

## 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 $G$ occurs infinitely often as a Galois group over the rationals $\mathbb Q$ with all nontrivial inertia groups of order $2$. Notably any such realization of $G$ can be translated up to a quadratic field over which the corresponding realization of $G$ is unramified. The sufficient conditions are imposed on a parametric polynomial with Galois group $G$--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 $A_5$, $PSL_2(7)$ and $PSL_3(3)$. Finally, the applications to $A_5$ and $PSL_3(3)$ are used to prove the existence of infinitely many optimally intersective realizations of these groups over the rational numbers (proved earlier for $PSL_2(7)$ by the first author).