# Trace

2010 Mathematics Subject Classification: Primary: 12F [MSN][ZBL]

The mapping $\mathrm{Tr}_{K/k}$ of a field $K$ into a field $k$ (where $K$ is a finite extension of $k$) that sends an element $\alpha \in K$ to the trace of the matrix (cf. Trace of a square matrix) of the $k$-linear mapping $K \rightarrow K$ sending $\beta \in K$ to $\alpha \beta$. $\mathrm{Tr}_{K/k}$ is a homomorphism of the additive groups $K^+ \rightarrow k^+$.

If $K/k$ is a separable extension, then $$\mathrm{Tr}_{K/k}(\alpha) = \sum_i \sigma_i(\alpha)$$ where the sum is taken over all $k$-isomorphisms $\sigma_i$ of $K$ into an algebraic closure $\bar k$ of $k$. The trace mapping is transitive, that is, if $L/K$ and $K/k$ are finite extensions, then for any $\alpha \in L$, $$\mathrm{Tr}_{L/k}(\alpha) = \mathrm{Tr}_{K/k}(\mathrm{Tr}_{L/K}(\alpha)) \ .$$

Especially in older mathematical literature, instead of $\mathrm{Tr}_{K/k}$ one finds $\mathrm{Sp}_{K/k}$ (from the German "Spur" ) as notation for this mapping.