Is there a nontrivial but short example where the separable degree is strictly less than the degree of a field extension?

Question

I consider the following theorem:

Let $E\supset F\supset k$ be a tower. Then $$[E:k]_s=[E:F]_s\cdot [F:k]_s.$$ Furthermore if $E$ is finite over $k$ then $[E:k]_s$ is finite and $$[E:k]_s\leq [E:k].$$

Now I am thinking about a nontrivial example where $$[E:k]_s<[E:k]$$ is strictly less.

But at the moment I can’t think about one. So I thought maybe one of you have an example in mind which could be helpful.

Answer 1

Litterally any extension which is not separable. It has to happen in characteristic $p>0$, and the standard example is: $k=\mathbb{F}_p(X)$ and $E=\mathbb{F}_p(\sqrt[p]{X})$ (adjoining a $p$th root of $X$).