POTW Semisimple Tensor Product of Fields

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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##.
 
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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.
 
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