Lemma 9.20.4. Let $L/K$ be a finite extension of fields. Let $V$ be a finite dimensional vector space over $L$. Let $\varphi : V \to V$ be an $L$-linear map. Then

$\text{Trace}_ K(\varphi : V \to V) = \text{Trace}_{L/K}(\text{Trace}_ L(\varphi : V \to V))$

and

$\det \nolimits _ K(\varphi : V \to V) = \text{Norm}_{L/K}(\det \nolimits _ L(\varphi : V \to V))$

Proof. Choose an isomorphism $V = L^{\oplus n}$ so that $\varphi$ corresponds to an $n \times n$ matrix. In the case of traces, both sides of the formula are additive in $\varphi$. Hence we can assume that $\varphi$ corresponds to the matrix with exactly one nonzero entry in the $(i, j)$ spot. In this case a direct computation shows both sides are equal.

In the case of norms both sides are zero if $\varphi$ has a nonzero kernel. Hence we may assume $\varphi$ corresponds to an element of $\text{GL}_ n(L)$. Both sides of the formula are multiplicative in $\varphi$. Since every element of $\text{GL}_ n(L)$ is a product of elementary matrices we may assume that $\varphi$ either looks like

$E_{12}(\lambda ) = \left( \begin{matrix} 1 & \lambda & \ldots \\ 0 & 1 & \ldots \\ \ldots & \ldots & \ldots \end{matrix} \right) \quad \text{or}\quad E_1(a) = \left( \begin{matrix} a & 0 & \ldots \\ 0 & 1 & \ldots \\ \ldots & \ldots & \ldots \end{matrix} \right)$

(because we may also permute the basis elements if we like). In both cases the formula is easy to verify by direct computation. $\square$

Comment #1771 by Carl on

K/L should probably be L/K on the RHS of the statement of the lemma.

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