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# 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 $T(v):=xv$.

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