À propos d’une version faible du problème inverse de Galois
Volume 197 / 2021
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 the paper’s 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 non-trivial finite group $G$, there exists a field 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.
