On parametric extensions over number fields

TL;DR

This paper investigates the existence of non-parametric Galois extensions over number fields, contributing to inverse Galois theory by establishing conditions for their existence based on the Galois group and regular extensions.

Contribution

It demonstrates that for any non-trivial finite group G and number field F, a non-G-parametric extension exists if G appears as a Galois group of some regular extension over F.

Findings

Existence of non-G-parametric extensions under certain conditions

Extension construction based on Galois groups and regularity

Relevance to inverse Galois problem and Beckmann-Black problem

Abstract

Given a number field $F$, a finite group $G$ and an indeterminate $T$, {\it{a $G$-parametric extension over $F$}} is a finite Galois extension $E/F(T)$ with Galois group $G$ and $E/F$ regular that has all the Galois extensions of $F$ with Galois group $G$ among its specializations. We are mainly interested in producing non-$G$-parametric extensions, which relates to classical questions in inverse Galois theory like the Beckmann-Black problem. Building on a strategy developed in previous papers, we show that there exists at least one non-$G$-parametric extension over $F$ for a given non-trivial finite group $G$ and a given number field $F$ under the sole necessary condition that $G$ occurs as the Galois group of a Galois extension $E/F(T)$ with $E/F$ regular.