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Tag 0BIH

Chapter 9: Fields > Section 9.20: Trace and norm

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 2320–2331 (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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