On the Euclidean minimum of some real number fields

Eva Bayer-Fluckiger[1]; Gabriele Nebe[2]

  • [1] Département de Mathématiques EPF Lausanne 1015 Lausanne Switzerland
  • [2] Lehrstuhl D für Mathematik RWTH Aachen 52056 Aachen Germany

Journal de Théorie des Nombres de Bordeaux (2005)

  • Volume: 17, Issue: 2, page 437-454
  • ISSN: 1246-7405

Abstract

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General methods from [3] are applied to give good upper bounds on the Euclidean minimum of real quadratic fields and totally real cyclotomic fields of prime power discriminant.

How to cite

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Bayer-Fluckiger, Eva, and Nebe, Gabriele. "On the Euclidean minimum of some real number fields." Journal de Théorie des Nombres de Bordeaux 17.2 (2005): 437-454. <http://eudml.org/doc/249464>.

@article{Bayer2005,
abstract = {General methods from [3] are applied to give good upper bounds on the Euclidean minimum of real quadratic fields and totally real cyclotomic fields of prime power discriminant.},
affiliation = {Département de Mathématiques EPF Lausanne 1015 Lausanne Switzerland; Lehrstuhl D für Mathematik RWTH Aachen 52056 Aachen Germany},
author = {Bayer-Fluckiger, Eva, Nebe, Gabriele},
journal = {Journal de Théorie des Nombres de Bordeaux},
keywords = {Euclidean rings; thin fields; lattices; Minkowski's conjecture; real quadratic fields; cyclotomic fields},
language = {eng},
number = {2},
pages = {437-454},
publisher = {Université Bordeaux 1},
title = {On the Euclidean minimum of some real number fields},
url = {http://eudml.org/doc/249464},
volume = {17},
year = {2005},
}

TY - JOUR
AU - Bayer-Fluckiger, Eva
AU - Nebe, Gabriele
TI - On the Euclidean minimum of some real number fields
JO - Journal de Théorie des Nombres de Bordeaux
PY - 2005
PB - Université Bordeaux 1
VL - 17
IS - 2
SP - 437
EP - 454
AB - General methods from [3] are applied to give good upper bounds on the Euclidean minimum of real quadratic fields and totally real cyclotomic fields of prime power discriminant.
LA - eng
KW - Euclidean rings; thin fields; lattices; Minkowski's conjecture; real quadratic fields; cyclotomic fields
UR - http://eudml.org/doc/249464
ER -

References

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  1. E. Bayer-Fluckiger, Lattices and number fields. Contemp. Math. 241 (1999), 69–84. Zbl0951.11016MR1718137
  2. E. Bayer-Fluckiger, Ideal lattices. A panorama of number theory or the view from Baker’s garden (Zürich, 1999), 168–184, Cambridge Univ. Press, Cambridge, 2002. Zbl1043.11057
  3. E. Bayer-Fluckiger, Upper bounds for Euclidean minima. J. Number Theory (to appear). Zbl1130.11066MR2274907
  4. J.W.S. Cassels, An introduction to the geometry of numbers. Springer Grundlehren 99 (1971). Zbl0209.34401MR306130
  5. J.H. Conway, N.J.A. Sloane, Low Dimensional Lattices VI: Voronoi Reduction of Three-Dimensional Lattices. Proc. Royal Soc. London, Series A 436 (1992), 55–68. Zbl0747.11027MR1177121
  6. J.H. Conway, N.J.A. Sloane, Sphere packings, lattices and groups. Springer Grundlehren 290 (1988). Zbl0634.52002MR920369
  7. P.M. Gruber, C.G. Lekkerkerker, Geometry of Numbers. North Holland (second edition, 1987) Zbl0611.10017MR893813
  8. The KANT Database of fields. http://www.math.tu-berlin.de/cgi-bin/kant/database.cgi. 
  9. F. Lemmermeyer, The Euclidean algorithm in algebraic number fields. Expo. Math. 13 (1995), 385–416. (updated version available via http://public.csusm.edu/public/FranzL/publ.html). Zbl0843.11046MR1362867
  10. C.T. McMullen, Minkowski’s conjecture, well-rounded lattices and topological dimension., Journal of the American Mathematical Society 18 (3) (2005), 711–734. Zbl1132.11034
  11. R. Quême, A computer algorithm for finding new euclidean number fields. J. Théorie de Nombres de Bordeaux 10 (1998), 33–48. Zbl0913.11056MR1827284
  12. E. Weiss, Algebraic number theory. McGraw-Hill Book Company (1963). Zbl0115.03601MR159805
  13. M. Dutour, A. Schürmann, F. Vallentin, A Generalization of Voronoi’s Reduction Theory and Applications, (preprint 2005). Zbl1186.11040

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