Trace for (finite) ring extensions. #
Suppose we have an
S with a finite basis. For each
s : S,
the trace of the linear map given by multiplying by
s gives information about
the roots of the minimal polynomial of
Implementation notes #
Typically, the trace is defined specifically for finite field extensions. The definition is as general as possible and the assumption that we have fields or that the extension is finite is added to the lemmas as needed.
x is in the base field
K, then the trace is
[L : K] * x.
L is not finite-dimensional over
x y : S to the trace of
x * y.
It is a symmetric bilinear form and is nondegenerate if the extension is separable.