# 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 $${H\cap K}$$.

## Article Details

How to Cite
Xuea, L. (2018). On galois extension of a separable algebra. International Journal of Algebra and Statistics, 7(1-2), 40-45. https://doi.org/10.20454/ijas.2018.1445
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