Definition:Trace (Field Theory)

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

Then by Field Extension is Vector Space, $L$ is naturally a vector space over $K$.

Let $\alpha \in L$, and $\theta_\alpha$ be the linear map:


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

The trace $N_{L/K} \left({\alpha}\right)$ of $\alpha$ is the trace of this map.