Relative Galois module structure of integers of abelian fields

Nigel P. Byott; Günter Lettl

Journal de théorie des nombres de Bordeaux (1996)

  • Volume: 8, Issue: 1, page 125-141
  • ISSN: 1246-7405

Abstract

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Let L / K be an extension of algebraic number fields, where L is abelian over ℚ . In this paper we give an explicit description of the associated order 𝒜 L / K of this extension when K is a cyclotomic field, and prove that o L , the ring of integers of L , is then isomorphic to 𝒜 L / K . This generalizes previous results of Leopoldt, Chan Lim and Bley. Furthermore we show that 𝒜 L / K is the maximal order if L / K is a cyclic and totally wildly ramified extension which is linearly disjoint to ℚ ( m ' ) / K , where m ' is the conductor of K .

How to cite

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Byott, Nigel P., and Lettl, Günter. "Relative Galois module structure of integers of abelian fields." Journal de théorie des nombres de Bordeaux 8.1 (1996): 125-141. <http://eudml.org/doc/247842>.

@article{Byott1996,
abstract = {Let $L/K$ be an extension of algebraic number fields, where $L$ is abelian over $\mathbb \{Q\}$. In this paper we give an explicit description of the associated order $\mathcal \{A\}_\{L/K\}$ of this extension when $K$ is a cyclotomic field, and prove that $o_L$, the ring of integers of $L$, is then isomorphic to $\mathcal \{A\}_\{L/K\}$. This generalizes previous results of Leopoldt, Chan Lim and Bley. Furthermore we show that $\mathcal \{A\}_\{L/K\}$ is the maximal order if $L/K$ is a cyclic and totally wildly ramified extension which is linearly disjoint to $\mathbb \{Q\}^\{(m^\{\prime \})\}/K$, where $m^\{\prime \}$ is the conductor of $K$.},
author = {Byott, Nigel P., Lettl, Günter},
journal = {Journal de théorie des nombres de Bordeaux},
keywords = {relative Galois module structure; abelian extension of a cyclotomic field; order},
language = {eng},
number = {1},
pages = {125-141},
publisher = {Université Bordeaux I},
title = {Relative Galois module structure of integers of abelian fields},
url = {http://eudml.org/doc/247842},
volume = {8},
year = {1996},
}

TY - JOUR
AU - Byott, Nigel P.
AU - Lettl, Günter
TI - Relative Galois module structure of integers of abelian fields
JO - Journal de théorie des nombres de Bordeaux
PY - 1996
PB - Université Bordeaux I
VL - 8
IS - 1
SP - 125
EP - 141
AB - Let $L/K$ be an extension of algebraic number fields, where $L$ is abelian over $\mathbb {Q}$. In this paper we give an explicit description of the associated order $\mathcal {A}_{L/K}$ of this extension when $K$ is a cyclotomic field, and prove that $o_L$, the ring of integers of $L$, is then isomorphic to $\mathcal {A}_{L/K}$. This generalizes previous results of Leopoldt, Chan Lim and Bley. Furthermore we show that $\mathcal {A}_{L/K}$ is the maximal order if $L/K$ is a cyclic and totally wildly ramified extension which is linearly disjoint to $\mathbb {Q}^{(m^{\prime })}/K$, where $m^{\prime }$ is the conductor of $K$.
LA - eng
KW - relative Galois module structure; abelian extension of a cyclotomic field; order
UR - http://eudml.org/doc/247842
ER -

References

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  1. [1] W. Bley, A Leopoldt-type result for rings of integers of cyclotomic extensions, Canad. Math. Bull.38 (1995), 141 - 148. Zbl0830.11037MR1335090
  2. [2] J. Brinkhuis, Normal integral bases and complex conjugation, J reine angew. Math.375/376 (1987), 157 - 166. Zbl0609.12009MR882295
  3. [3] S.-P. Chan & C.-H. Lim, Relative Galois module structure of rings of integers of cyclotomic fields, J. reine angew. Math.434 (1993), 205 -220. Zbl0753.11038MR1195696
  4. [4] A. Fröhlich, Galois module structure of algebraic integers, Erg. d. Math.3, vol. 1, Springer, 1983. Zbl0501.12012MR717033
  5. [5] A. Fröhlich & M.J. Taylor, Algebraic number theory, Camb. Studies Adv. Math. vol. 27, Cambridge University Press, 1991. Zbl0744.11001
  6. [6] H.-W. Leopoldt, Über die Hauptordnung der ganzen Elemente eines abelschen Zahlkörpers, J. reine angew. Math.201 (1959), 119 -149. Zbl0098.03403MR108479
  7. [7] G. Lettl, The ring of integers of an abelian number field, J. reine angew. Math.404 (1990), 162-170. Zbl0703.11060MR1037435
  8. [8] G. Lettl, Note on the Galois module structure of quadratic extensions, Coll. Math.67 (1994), 15 - 19. Zbl0812.11063MR1292939
  9. [9] K.W. Roggenkamp & M.J. Taylor, Group rings and class groups, DMV-Seminar Bd. 18, Birkhäuser, 1992. Zbl0742.00085MR1167449
  10. [10] M.J. Taylor, Relative Galois module structure of rings of integers, Orders and their applications (Proceedings of Oberwolfach 1984) (I. Reiner & K.W. Roggenkamp, eds.), Lect. Notes1142, Springer, 1985, pp. 289-306. Zbl0578.12004MR812505

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