Definition:Field Norm/Definition 2

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

Let $\alpha\in L$. 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 $\ds \prod_{\sigma \mathop \in \map {\operatorname{Gal}} {L / K}} \map \sigma \alpha$.