# Definition:Field Norm

 It has been suggested that this page or section be renamed: Perhaps "Extension Norm", for "Field Norm" is ambiguous with Definition:Norm on Division Ring One may discuss this suggestion on the talk page.

## Definition

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 $\operatorname{Gal} \left({L / K}\right)$ is finite.

The field norm $N_{L / K} \left({\alpha}\right)$ of $\alpha$ is $\displaystyle \prod_{\sigma \mathop \in \operatorname{Gal} \left({L / K}\right)} \sigma \left({\alpha}\right)$.