I've been struggling with this question for some days now. Let $K$ be a field extension of $k$, and $x$ an invertible linear transformation of the $K$-vector space $V$. If we consider $V$ as a $k$-vector space and $x$ as $k$-linear then $$ det_kx = N(det_Kx) $$ where $det_Kx$ denotes the determinant of $K$ viewed as $K$-linear (same for $k$). This in fact generalizes the definition of field norm.

I have showed this is true for diagonal matrices, but not in general. In fact, I'm not sure this is the right equality, but I think so. I hope someone can give me a hint (or a reference :) I would be most thankful.

Greetings.

J.

P.S. In my case $K=\mathbb{F}_{q^2}$ and $k=\mathbb{F}_q$