R\'ealisations galoisiennes explicites de certaines familles de $2$-groupes

TL;DR

This paper constructs explicit Galois extensions over rational function fields for certain 2-groups and identifies specializations that yield Galois groups over the rationals, advancing explicit realization techniques.

Contribution

It provides explicit constructions of Galois extensions with specified 2-group Galois groups and identifies arithmetic progressions for specializations over .

Findings

Explicit Galois extensions for certain 2-groups over (T)

Arithmetic progressions of integers with prescribed Galois groups

Method for realizing specific 2-groups as Galois groups over 

Abstract

In this paper, we construct, for some $2$-groups $G$, explicit Galois extensions $E/\mathbb{Q}(T)$ of group $G$ with $E\cap\overline{\mathbb{Q}}=\mathbb{Q}$. We also provide explicit arithmetic progressions of integers $t_0$ such that the specialization $E_{t_0}/\mathbb{Q}$ of $E/\mathbb{Q}(T)$ at $t_0$ has Galois group $G$.