Field trace

Summary

Let $L/K$ be a finite extension and let $\alpha \in L$. Multiplication by $\alpha$ is a $K$-linear map of $L$, and the field trace $\mathrm{Tr}_{L/K}(\alpha)$ is its trace. It is a $K$-linear map $L \to K$. For a separable extension it is the sum of the images of $\alpha$ under all embeddings of $L$ into an algebraic closure. The trace gives the trace form and the discriminant of an extension. The trace map is not identically zero exactly when the extension is separable.