  Let L/K be an extension of absolutely abelian number fields of equal
conductor, n. The image of the ring of integers of L under the trace map from L
to K is an ideal in the ring of integers in K. We compute the absolute norm of
this ideal exactly for any such L/K, thereby sharpening an earlier result of
Kurt Girstmair. Furthermore, we define an "adjusted trace map" that allows the
proof of Leopoldt's Theorem to be reduced to the cyclotomic case.
