A propos d'une version faible du probl\`eme inverse de Galois

TL;DR

This paper explores the Weak Inverse Galois Problem, demonstrating how to generate fields satisfying it without fulfilling the classical problem, and shows that every field satisfies its regular version.

Contribution

It provides methods to produce fields fulfilling the Weak Inverse Galois Problem that do not satisfy the classical version, expanding understanding of field extensions and Galois groups.

Findings

Fields like $ ext{Q}^{ ext{sol}}$, $ ext{Q}^{ ext{tr}}$, $ ext{Q}^{ ext{pyth}}$ satisfy the problem.

For every non-trivial finite group G, many fields fulfill the problem without Galois group G.

All fields satisfy the regular version of the Weak Inverse Galois Problem.

Abstract

This paper deals with the Weak Inverse Galois Problem which, for a given field $k$, states that, for every finite group $G$, there exists a finite separable extension $L/k$ such that ${\rm{Aut}}(L/k)=G$. One of its goals is to explain how one can generically produce families of fields which fulfill this problem, but which do not fulfill the usual Inverse Galois Problem. We show that this holds for, e.g., the fields $\mathbb{Q}^{{{\rm{sol}}}}$, $\mathbb{Q}^{{{\rm{tr}}}}$, $\mathbb{Q}^{{{\rm{pyth}}}}$, and for the maximal pro-$p$-extensions of $\mathbb{Q}$. Moreover, we show that, for every finite non-trivial group $G$, there exists many fields fulfilling the Weak Inverse Galois Problem, but over which $G$ does not occur as a Galois group. As a further application, we show that every field fulfills the regular version of the Weak Inverse Galois Problem.