Associativity of norms in inseparable extensions

Question

Let $K$ be a field. Let $L/K$ and $E/L$ be finite extensions. Let $α$ be an element of E. Let $N_{E/K}(α)$ be the norm of $α$, i.e. the determinant of the regular representaion matrix of $α$. It is well known that $N_{E/K}(α)$ $=$ $N_{L/K}(N_{E/L}(α))$ if $E/K$ is separable. I tried to find a proof of this formula in inseparable extensions, but failed. Where can I find it? It'd be also nice if someone provides a sketch of the proof, here.

Answer 1

The proof can be found here.