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Let  be a Galois extension, and let  . The trace
 of  is defined to be the sum of all the elements of the orbit of  under the group action of the Galois group
 on  ; taken with multiplicities if  is a finite extension.
In the case where  is a finite extension,
The trace of  is always an element of  , since any element of
 permutes the orbit of  and thus fixes
 .
The name “trace” derives from the fact that, when  is finite, the trace of  is simply the trace of the linear transformation
 of vector spaces over  defined by
 .
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