I have a problem,
Let $k$ be a field of characteristic $p>0$, let $L=k(X,Y)$ and $K=k(X^p,Y^p)$. It's evident that $L/K$ is finite and $[ L:K ]=p^2$. Suppose $L/K$ is simple then $L=K(\alpha)$, then $min(K,\alpha)$ must be of degree $p^2$, however $min(K,\alpha)|(x^p-\alpha^p)$, which leads to a contradiction.
So $L/K$ is not simple, we deduce from Primitive Element Theorem that there are infinitely middle fields of $L/K$.
Here comes the part I couldn't solve, show EXPLICIT INFINITELY MANY MIDDLE FIELDS of $L/K$.
I'm stuck because, let $k=F_p$, then if there's no mistaken, there's only finitely many simple extension of $K$, and I coundn't imagine a bigger extension. So help me with this, thank you. show EXPLICIT INFINITELY MANY MIDDLE FIELDS of $L/K$.