On Galois extension of a separable algebra

Abstract

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 a subgroup \(H \) of \(G \), \(B\supset B^H \) is a Hirata separable Galois extension with Galois group \(H\) if and only if \(H\subset K \) where \(K=\{g\in G\,|\,g(c)=c \) for each \(c\in C\} \). Moreover, for \(H\not\subset K \) and \(H\cap K\not=\{e\} \), it is shown that \(B^{H\cap K}\supset B^H \) is the unique minimal Galois extension with Galois group \(H/H\cap K\) in \(B\supset B^H \) such that \(B\supset B^{H\cap K} \) is a Hirata separable Galois extension with Galois group \({H\cap K}\).