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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.

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up vote 2 down vote accepted

The proof can be found here.

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Thanks. That's a nice proof. –  Makoto Kato May 8 '12 at 2:06
    
By the way, I think the proof of Lemma 1.1 in the above link is incorrect. –  Makoto Kato May 8 '12 at 2:38
    
I found another proof. See my comments here(math.stackexchange.com/questions/50737/… ). –  Makoto Kato May 14 '12 at 0:58
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