- Field trace
In
mathematics , the field trace is alinear mapping defined for certainfield extension s. If "L"/"K" is a finiteGalois extension , it is defined for α in "L" as the sum of all the conjugates:"g"(α)
of α, for "g" in the
Galois group "G" of "L" over "K". It is a "K"-linear map of "L" to "K", written:Tr"L"/"K".
It is often used as a
quadratic form , particularly inalgebraic number theory and the theory of thedifferent ideal , in the shape:<α,β> → Tr"L"/"K"(αβ).
The connection with the trace of a square matrix can be explained by means of the multiplication action of α on "L", considered as a "K"-linear mapping. This leads to a more general definition.
If the powers of α span "L" as "K"-vector space, it is easy to write down the matrix of α (the
companion matrix ) and so compute the trace. It is the negative of the ("n" − 1)-th coefficient of theminimal polynomial for the matrix, where "n" = ["L": "K"] , and so the sum of its roots. When "L" is a Galois extension of "K" it follows that the matrix for multiplication by α actually diagonalises over "L", with eigenvalues the "g"(α).That was all under the simplifying assumption that the powers of α span "L". The general situation is that they span a proper subfield "M" = "K"(α) - in that case the same argument can be applied to a
direct sum of "M"-invariant subspaces.The conclusion is that the field trace defined by use of the Galois group is a special case of the trace of the multiplication action, which is available for any finite extension, Galois or not.
See also
*
Field norm
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