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Let $F$ be a perfect field, if there exists some finite separable extension field $E$ of $F$ such that $E=F\oplus \cdots \oplus F$, where $E$ can be seen as a separable $F$-algebra.

In fact I want to prove that $M_n(E)=E\otimes M_n(F)=M_n(F)\oplus \cdots \oplus M_n(F)$...

Thanks very much!

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    what is the product on the right hand side of the equality? It s always true is you consider the $F$-vector structure of $E$. – 2017-01-10
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    $E=F$ works just fine. I suspect you mean something else. – 2017-01-10
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    Let $F$ be a perfect field, then $E$ is a separable $F$-algebra......I want to know that if $E$, as a $F$-algebra, is a direct sum of some $F$? – 2017-01-10

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