# The Stacks Project

## Tag 0BIH

Lemma 9.20.3. Let $L/K$ be a finite extension of fields. Let $\alpha \in L$ and let $P = x^d + a_1 x^{d - 1} + \ldots + a_d$ be the minimal polynomial of $\alpha$ over $K$. Then $$\text{Norm}_{L/K}(\alpha) = (-1)^{[L : K]} a_d^e \quad\text{and}\quad \text{Trace}_{L/K}(\alpha) = - e a_1$$ where $e d = [L : K]$.

Proof. Follows immediately from Lemma 9.20.2 and the definitions. $\square$

The code snippet corresponding to this tag is a part of the file fields.tex and is located in lines 2321–2332 (see updates for more information).

\begin{lemma}
\label{lemma-trace-and-norm-from-minimal-polynomial}
Let $L/K$ be a finite extension of fields. Let $\alpha \in L$ and let
$P = x^d + a_1 x^{d - 1} + \ldots + a_d$
be the minimal polynomial of $\alpha$ over $K$. Then
$$\text{Norm}_{L/K}(\alpha) = (-1)^{[L : K]} a_d^e \quad\text{and}\quad \text{Trace}_{L/K}(\alpha) = - e a_1$$
where $e d = [L : K]$.
\end{lemma}

\begin{proof}
Follows immediately from Lemma \ref{lemma-characteristic-vs-minimal-polynomial}
and the definitions.
\end{proof}

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