Let K, L be algebraic number fields with K ⊆ L, and , their respective rings of integers. We consider the trace map and the -ideal . By I(L/K) we denote the group indexof in (i.e., the norm of over ℚ). It seems to be difficult to determine I(L/K) in the general case. If K and L are absolutely abelian number fields, however, we obtain a fairly explicit description of the number I(L/K). This is a consequence of our description of the Galois module structure of (Theorem 1). The case of equal conductors of the fields K, L is of particular interest. Here we show that I(L/K) is a certain power of 2 (Theorems 2, 3, 4).
@article{bwmeta1.element.bwnjournal-article-aav62i4p383bwm,
author = {Kurt Girstmair},
title = {On the trace of the ring of integers of an abelian number field},
journal = {Acta Arithmetica},
volume = {62},
year = {1992},
pages = {383-389},
zbl = {0739.11051},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-aav62i4p383bwm}
}
Kurt Girstmair. On the trace of the ring of integers of an abelian number field. Acta Arithmetica, Tome 62 (1992) pp. 383-389. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-aav62i4p383bwm/
[000] [1] K. Girstmair, Dirichlet convolution of cotangent numbers and relative class number formulas, Monatsh. Math. 110 (1990), 231-256. | Zbl 0717.11048
[001] [2] K. Ireland and M. Rosen, A Classical Introduction to Modern Number Theory, Springer, New York 1982. | Zbl 0482.10001
[002] [3] H. W. Leopoldt, Über die Hauptordnung der ganzen Elemente eines abelschen Zahlkörpers, J. Reine Angew. Math. 201 (1959), 119-149. | Zbl 0098.03403
[003] [4] G. Lettl, The ring of integers of an abelian number field, J. Reine Angew. Math. 404 (1990), 162-170. | Zbl 0703.11060