The Stacks project

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

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


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

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. $\square$

Comments (1)

Comment #1771 by Carl on

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

There are also:

  • 1 comment(s) on Section 9.20: Trace and norm

Post a comment

Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.

In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 0BII. Beware of the difference between the letter 'O' and the digit '0'.