Examples of finite-dimensional, semisimple, non-separable $k$-algebras

Question

1. Motivation
Let $k$ be a field.
Apparently, any separable $k$-algebra is finite-dimensional and semisimple.
Using Maschke's theorem for Hopf algebras, one can prove that for Hopf algebras the following stronger statement holds:

A $k$-Hopf algebra is separable if and only if it is finite-dimensional and semisimple.

I believe, if you replace "$k$-Hopf algebra" by "$k$-algebra", this proposition is wrong.

2. Question

• What are examples of finite-dimensional, semisimple, non-separable algebras?

Apparently, if $k$ is a perfect field any finite-dimensional, semisimple $k$-algebra is separable. Hence, any example has to be over an imperfect field.

Answer 1

For any prime $p$, the $\mathbb{F}_p(T)$-algebra (which is simple since it is a field) $\mathbb{F}_p(T)(\sqrt[p]{T})$ is such an example.