Let $K/F$ be a Galois extension, and let $x \in K$ . The trace $\operatorname{Tr}_F^K(x)$ of $x$ is defined to be the sum of all the elements of the orbit of $x$ under the group action of the Galois group $\operatorname{Gal}(K/F)$ on $K$ ; taken with multiplicities if $K/F$ is a finite extension.

In the case where $K/F$ is a finite extension, $$ \operatorname{Tr}_F^K(x) := \sum_{\sigma \in \operatorname{Gal}(K/F)} \sigma(x) $$

The trace of $x$ is always an element of $F$ , since any element of $\operatorname{Gal}(K/F)$ permutes the orbit of $x$ and thus fixes $\operatorname{Tr}_F^K(x)$ .

The name ``trace'' derives from the fact that, when $K/F$ is finite, the trace of $x$ is simply the trace of the linear transformation $T: K \longrightarrow K$ of vector spaces over $F$ defined by $T(v) := xv$ .