Definition:Field Norm

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Let $K$ be a field and $L / K$ a finite field extension of $K$.

Let $\alpha\in L$.

Definition 1

By Vector Space on Field Extension is Vector Space, $L$ is naturally a finite dimensional vector space over $K$.

Let $\theta_\alpha$ be the linear operator:

$\theta_\alpha: L \to L: \beta \mapsto \alpha \beta$

The field norm $\map {N_{L/K} }\alpha$ of $\alpha$ is the determinant of $\theta_\alpha$.

Definition 2: for Galois extensions

Let $L / K$ be Galois.

By Finite Field Extension has Finite Galois Group, the Galois group $\map {\operatorname{Gal}} {L / K}$ is finite.

The field norm $\map {N_{L / K}} \alpha$ of $\alpha$ is $\ds \prod_{\sigma \mathop \in \map {\operatorname{Gal}} {L / K}} \map \sigma \alpha$.

Also see