For (a): a field extension $L/k$ is separable is $L$ is a separable algebra, that is, if $L\otimes K$ is a semisimple algebra for all field extensions $K/k$. In particular, if $L$ is separable over $k$ and $K$ is an extension of $k$, then $L\otimes K$ will have no nilpotent elements because it is semisimple.

For details, see for example, Pierce's beautiful book Associative algebras.

For (b), for which the argument is more elaborate, see Jacobson's *Lectures in Abstract Algebra*, vol. 3, where the result is part 3 of Theorem 21; he does without algebraicity, by the way.