On the trace map between absolutely abelian number fields of equal conductor

Abstract

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.

11 pages, 1 figure, uses xypic and amscd. Completely revised version. To appear in Acta Arithmetica