GALOIS EMBEDDING PROBLEMS WITH CYCLIC QUOTIENT OF ORDER p

JAN MINAC JOHN SWALLOW

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