\(\mathrm{GL}_n (q)\) as Galois group over the rationals
We prove a new criterion for realizing groups as Galois groups over the rationals. This generalizes Thompson's rigidity criterion, giving weaker conditions that include the action of the braid group on certain generating systems. Using this criterion, most of the groups \(\mathrm{GL}_n(q)\) with \(n\) even and \(n\geq 3(q-1)\) are realized as Galois groups of regular extensions of \(\mathbb{Q}(x)\). The Chevalley groups that had been realized before in this way comprised only groups over fields of prime order \(p\), or order \(p^2\).
scientific article; zbMATH DE number 4216042
scientific article