The inverse problem of Galois theory: the rigidity method

Abstract

[en] The development of Galois theory was a key turning point in the history of mathematics. It began the study of fields that are still active and solved one of the most important problems in mathematics at the time.

The Inverse Galois Problem asks whether given a finite group $G$ and a field $K,$ if it is possible to find a Galois extension $L / K$ such that $G \cong \operatorname{Gal}(L / K) .$ The answer to this problem depends, of course, on the properties of the group $G$ and on the properties of the field $K .$ For instance, the solution to this problem is positive only for cyclic groups when $K$ is a finite field, whereas the solution is always positive when $K=\mathbb{C}(t)$. It remains an open problem to show whether all groups are Galois groups over $K=\mathbb{Q}$, although some partial solutions have been given. For example, in 1937 Scholz $[\mathrm{Sch} 37]$ and Reichardt $[\mathrm{Rei} 37]$ simultaneously but independently proved that $p$ -groups can be realised as Galois groups over $\mathbb{Q}$ for any odd prime $p .$ Taking this as a starting point, later on Safarevic proved that all solvable groups are Galois groups over $\mathbb{Q}$ in $[\text { S } 54]$

This work will study one of the main methods developed to partially solve the problem, the Rigidity Method. These techniques first appeared in the work of Belyi [Bel79], Matzat $[\text { Mat } 84]$ and Thompson [Tho84c]. The Rigidity Method takes as starting point the solution in $\mathbb{C}(t)$.