Galois embedding problems with cyclic quotient of order p

Jan Minac and John Swallow

\newcommand{\Fp}{\mathbb{F}_p} \def\Gal{\mathrm{Gal}}% Let $K$ be a cyclic Galois extension of degree $p$ over a field $F$ containing a primitive $p$th root of unity. We consider Galois embedding problems involving Galois groups with common quotient $\Gal(K/F)$ such that corresponding normal subgroups are indecomposable $\Fp[\Gal(K/F)]$-modules. For these embedding problems we prove conditions on solvability, formulas for explicit construction, and results on automatic realizability.