trace

Let $K/F$ be a Galois extension, and let $x\in K$. The trace ${\mathrm{Tr}}_F^K(x)$ 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 $\mathrm{Gal}(K/F)$ on $K$; taken with multiplicities if $K/F$ is a finite extension.

In the case where $K/F$ is a finite extension,

$${\mathrm{Tr}}_F^K(x):=\sum _{\sigma \in \mathrm{Gal}(K/F)}\sigma (x)$$

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

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 $T:K⟶K$ of vector spaces over $F$ defined by $T(v):=xv$.