# On the trace of the ring of integers of an abelian number field

Acta Arithmetica (1992)

- Volume: 62, Issue: 4, page 383-389
- ISSN: 0065-1036

## Access Full Article

top## Abstract

top## How to cite

topKurt Girstmair. "On the trace of the ring of integers of an abelian number field." Acta Arithmetica 62.4 (1992): 383-389. <http://eudml.org/doc/206500>.

@article{KurtGirstmair1992,

abstract = {Let K, L be algebraic number fields with K ⊆ L, and $O_K$, $O_L$ their respective rings of integers. We consider the trace map
$T = T_\{L/K\} : L → K$ and the $O_K$-ideal $T(O_L) ⊆ O_K$. By I(L/K) we denote the group indexof $T(O_L)$ in $O_K$ (i.e., the norm of $T(O_L)$ 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 $T(O_L)$ (Theorem 1). The case of equal conductors $f_K = f_L$ 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).
},

author = {Kurt Girstmair},

journal = {Acta Arithmetica},

keywords = {abelian number fields; rings of integers; trace map; Galois module structure; group index},

language = {eng},

number = {4},

pages = {383-389},

title = {On the trace of the ring of integers of an abelian number field},

url = {http://eudml.org/doc/206500},

volume = {62},

year = {1992},

}

TY - JOUR

AU - Kurt Girstmair

TI - On the trace of the ring of integers of an abelian number field

JO - Acta Arithmetica

PY - 1992

VL - 62

IS - 4

SP - 383

EP - 389

AB - Let K, L be algebraic number fields with K ⊆ L, and $O_K$, $O_L$ their respective rings of integers. We consider the trace map
$T = T_{L/K} : L → K$ and the $O_K$-ideal $T(O_L) ⊆ O_K$. By I(L/K) we denote the group indexof $T(O_L)$ in $O_K$ (i.e., the norm of $T(O_L)$ 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 $T(O_L)$ (Theorem 1). The case of equal conductors $f_K = f_L$ 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).

LA - eng

KW - abelian number fields; rings of integers; trace map; Galois module structure; group index

UR - http://eudml.org/doc/206500

ER -

## References

top- [1] K. Girstmair, Dirichlet convolution of cotangent numbers and relative class number formulas, Monatsh. Math. 110 (1990), 231-256. Zbl0717.11048
- [2] K. Ireland and M. Rosen, A Classical Introduction to Modern Number Theory, Springer, New York 1982. Zbl0482.10001
- [3] H. W. Leopoldt, Über die Hauptordnung der ganzen Elemente eines abelschen Zahlkörpers, J. Reine Angew. Math. 201 (1959), 119-149. Zbl0098.03403
- [4] G. Lettl, The ring of integers of an abelian number field, J. Reine Angew. Math. 404 (1990), 162-170. Zbl0703.11060

## NotesEmbed ?

topTo embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.