On compositions of a Galois extension of a separable algebra

Let B be a Galois extension of B G with Galois group G such that B G is a separable C G -algebra where C is the center of B. Then for every subgroup H of G, B is a composition of a Hirata separable Galois extension B of B H∩K with Galois group H ∩ K and a Galois extension B H∩K of B G with Galois group H/H ∩ K where K = {g ∈ G |g(c )= c for all c ∈ C}. This generalizes DeMeyer’s result for an indecomposable Galois algebra. Moreover, we show that B is a composition of a Hirata separable Galois extension B of B G C with Galois group K and a DeMeyer-Kanzaki Galois extension B G C of B G with Galois group G/K