A note on finite embedding problems with nilpotent kernel

2020/11/15 by Arno Fehm, Fehm, Arno, François Legrand +1 · 1 citation
Mathematics · Computer Science · #Algebraic Geometry and Number Theory #Coding theory and cryptography #Finite Group Theory Research

paper · pdf · doi:10.48550/arxiv.2011.07536

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 S of primes of K, every finite split embedding problem G → \rmGal(L/K) over K with nilpotent kernel has a solution \rmGal(F/K) → 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 σ of K the embedding problem acquires a solution over the skew field of fractions K(T, σ) of the twisted polynomial ring K[T, σ].

Cited by

• On finite embedding problems with abelian kernels

Related

• On finite embedding problems with abelian kernels
• Problèmes de plongement finis sur les corps non commutatifs
• Split embedding problems over the open arithmetic disc
• Embedding problems for automorphism groups of field extensions
• Parameterizing solutions to any Galois embedding problem over Z/pⁿZ with elementary p-abelian kernel