A note on finite embedding problems with nilpotent kernel

Abstract

The first aim of this note is to fill a gap in the literature by giving a proof of the following refinement of Shafarevich's theorem on solvable Galois groups: Given a global field $k$, a finite set $\mathcal{S}$ of primes of $k$, and a finite solvable group $G$, there is a Galois field extension of $k$ of Galois group $G$ in which all primes in $\mathcal{S}$ are totally split. To that end, we prove that, given a global field $k$ and a finite set $\mathcal{S}$ of primes of $k$, every finite split embedding problem $G \rightarrow {\rm{Gal}}(L/k)$ over $k$ with nilpotent kernel has a solution ${\rm{Gal}}(F/k) \rightarrow G$ such that all primes in $\mathcal{S}$ are totally split in $F/L$. We then use this to contribute to inverse Galois theory over division rings. Namely, 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 $k[T, \sigma]$.