A note on finite embedding problems with nilpotent kernel

Fehm, Arno; Legrand, François

Journal de théorie des nombres de Bordeaux, Volume 34 (2022) no. 2, pp. 549-562.

The first aim of this note is to fill a gap in the literature by proving that, given a global field $K$ and a finite set $𝒮$ of primes of $K$, every finite split embedding problem $G\to \mathrm{Gal}(L/K)$ over $K$ with nilpotent kernel has a solution $\mathrm{Gal}(F/K)\to G$ such that all primes in $𝒮$ are totally split in $F/L$. We then apply this to inverse Galois theory over division rings. Firstly, given a number field $K$ of level at least $4$, we show that every finite solvable group occurs as a Galois group over the division ring $H_K$ of quaternions with coefficients in $K$. Secondly, given a finite split embedding problem with nilpotent kernel over a finite field $K$, we fully describe for which automorphisms $\sigma$ of $K$ the embedding problem acquires a solution over the skew field of fractions $K(T,\sigma )$ of the twisted polynomial ring $K[T,\sigma ]$.

Le premier objectif de cet article est de combler une lacune de la littérature en montrant que, si $K$ est un corps global et si $𝒮$ est un ensemble fini de places de $K$, alors tout problème de plongement fini scindé $G\to \mathrm{Gal}(L/K)$ sur $K$ à noyau nilpotent admet une solution $\mathrm{Gal}(F/K)\to G$ telle que toutes les places dans $𝒮$ soient totalement décomposées dans $F/L$. Nous appliquons ensuite cela à la théorie inverse de Galois sur les corps non nécessairement commutatifs. Tout d’abord, étant donné un corps de nombres $K$ de niveau au moins 4, nous montrons que tout groupe fini résoluble est groupe de Galois sur le corps des quaternions $H_K$ à coefficients dans $K$. Ensuite, étant donné un problème de plongement fini scindé à noyau nilpotent sur un corps fini $K$, nous décrivons complètement les automorphismes $\sigma$ de $K$ pour lesquels le problème de plongement admet une solution sur le corps de fractions $K(T,\sigma )$ de l’anneau de polynômes $K[T,\sigma ]$.

Classification: 12F12, 12E30, 11R32, 12E15, 11R52
Keywords: Finite embedding problems, global fields, inverse Galois theory, division rings, quaternions

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