Definition:Field Norm/Definition 1

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

Let $\alpha\in L$. By 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 $N_{L/K} \left({\alpha}\right)$ of $\alpha$ is the determinant of $\theta_\alpha$.