Are normal extensions always simple?

Question

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?

Answer 1

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.