A note on finite embedding problems with nilpotent kernel

TL;DR

This paper proves that finite split embedding problems with nilpotent kernel over global fields have solutions with specified splitting conditions and applies these results to inverse Galois problems over division rings and skew fields.

Contribution

It fills a gap by proving solutions exist for split embedding problems with nilpotent kernel over global fields and explores applications to inverse Galois theory over division rings.

Findings

Solutions exist for split embedding problems with nilpotent kernel over global fields.

Every finite solvable group occurs as a Galois group over certain division rings.

Full description of solutions over skew fields of fractions of twisted polynomial rings.

Abstract

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 $\mathcal{S}$ 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 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 $K[T, \sigma]$.