Field trace - Definition


Field trace - Definition

In mathematics, the field trace is a linear mapping defined for certain field extensions. If L/K is a finite Galois 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 in algebraic number theory and the theory of the different 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 the minimal 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.