Definition:Trace (Field Theory)

This page is about Trace in the context of Field Theory. For other uses, see Trace.

Definition

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

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

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

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

The trace $\map {\operatorname {Tr}_{L / K} } \alpha$ of $\alpha$ is the trace of this linear operator.

Also see

• Definition:Field Norm

Sources

• 1973: Gerald J. Janusz: Algebraic Number Fields Chapter $\text{I}$: Subrings of Fields: $\S5$: Norms and Traces