Definition:Trace (Field Theory)
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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
Sources
- 1973: Gerald J. Janusz: Algebraic Number Fields Chapter $\text{I}$: Subrings of Fields: $\S5$: Norms and Traces