Let K/F be a Galois extensionMathworldPlanetmath, and let xK. The trace TrFK(x) of x is defined to be the sum of all the elements of the orbit of x under the group actionMathworldPlanetmath of the Galois groupMathworldPlanetmath Gal(K/F) on K; taken with multiplicities if K/F is a finite extensionMathworldPlanetmath.

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


The trace of x is always an element of F, since any element of Gal(K/F) permutes the orbit of x and thus fixes TrFK(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:KK of vector spacesMathworldPlanetmath over F defined by T(v):=xv.

Title trace
Canonical name Trace1
Date of creation 2013-03-22 12:17:59
Last modified on 2013-03-22 12:17:59
Owner djao (24)
Last modified by djao (24)
Numerical id 7
Author djao (24)
Entry type Definition
Classification msc 12F05