Are normal extensions always simple? I'm trying to diagram out the relations between simple, normal, separable, and Galois field extensions in my head. I understand that all separable extensions are simple, and Galois extensions are normal and separable. Then, not all separable extensions are normal, and vice versa.
I don't quite know how the Venn Diagram circles for normal extensions and simple extensions interact. There are simple extensions that aren't normal, but are there normal extensions that aren't simple?

There are normal inseparable extensions that are not simple. For instance,
let $L=\Bbb F_2(x,y)$ and $K=\Bbb F_2(x^2,y^2)$. Then $L/K$ is normal, inseparable and not simple.
