Galois embedding problems with cyclic quotient of order $p$
Jan Minac and John Swallow
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 ${\rm Gal}(K/F)$ such that corresponding normal subgroups are indecomposable ${{\mathbb F}_p}[{\rm Gal}(K/F)]$-modules. For these embedding problems we prove conditions on solvability, formulas for explicit construction, and results on automatic realizability.