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

where $e d = [L : K]$.

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.
$\square$

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