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

Kurt Girstmair

Acta Arithmetica (1992)

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

Abstract

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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).

How to cite

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Kurt 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

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

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