# trace

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 $T(v):=xv$.

Title trace Trace1 2013-03-22 12:17:59 2013-03-22 12:17:59 djao (24) djao (24) 7 djao (24) Definition msc 12F05