Field extension trace

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Definition

Suppose $L/K$ is a finite field extension. For $\alpha \in L$, define the map $m_{\alpha }:L\to L$ defined by $m_{\alpha }(x)=\alpha x$ .

Then the trace of $\alpha$ is defined as the trace (in the linear algebra sense) of the linear map $m_{\alpha }$.

This map is often denoted $\mathrm {tr} _{L/K}(\alpha )$.

See also

• Field extension norm - a related map, defined as the determinant instead of the trace of the map $m_{\alpha }$.