Semisimple Tensor Product of Fields

[Euge]

Let $L/k$ be a field extension. Suppose $F$ is a finite separable extension of $k$. Prove $L\otimes_k F$ is a semisimple algebra over $k$.

 

[Euge]

Spoiler

By the primitive element theorem, $F = k(\alpha)$ for some $\alpha\in F$ separable over $k$. Let $f(x)$ be the minimal polynomial of $\alpha$ over $k$. In $L[x]$, $f(x)$ is the product of distinct irreducibles $f_1(x),\ldots, f_d(x)$. Hence $$L\otimes_k F \simeq L\otimes_k \frac{k[x]}{(f(x))} \simeq \frac{L[x]}{(f(x))} \simeq \bigoplus_{j = 1}^d\frac{L[x]}{(f_j(x))}$$ is a direct sum of fields. Thus $L\otimes_k F$ is semisimple.