I am trying to prove the following homework problem:
Let $K/k$ be a normal field extension and $K_i$ and $K_s$ be intermediate extensions so that $K_i/k$ and $K/K_s$ are purely inseparable and $K_s/k$ and $K/K_i$ are separable. Show that $K=K_i\cdot{K_s}$ and $K_i \cap K_s=k$.
Showing $K_s \cap K_i =k$ is simple. Proving $K=K_i \cdot{K_s}$ seems a bit harder. My question is the following:
Is the condition $K=K_i \cdot K_s$ equivalent to showing $K_i\setminus k$ and $K_s \setminus k$ form a partition of $K\setminus k$?
I dont think this is correct, because I cannot see how the hypothesis of normality comes into play.