A computer algorithm for finding new euclidean number fields

Roland Quême

Journal de théorie des nombres de Bordeaux (1998)

  • Volume: 10, Issue: 1, page 33-48
  • ISSN: 1246-7405

Abstract

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This article describes a computer algorithm which exhibits a sufficient condition for a number field to be euclidean for the norm. In the survey [3] p 405, Franz Lemmermeyer pointed out that 743 number fields where known (march 1994) to be euclidean (the first one, ℚ , discovered by Euclid, three centuries B.C.!). In the first months of 1997, we found more than 1200 new euclidean number fields of degree 4, 5 and 6 with a computer algorithm involving classical lattice properties of the embedding of the degree n field 𝐊 into ℝ n and the structure of the unit group of 𝐊 . This articles ends with a generalization of the method for the determination of rings of S -integers of number fields euclidean for the norm and for the study of the inhomogeneous minimum of the norm form. Our results are in accordance with known results.

How to cite

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Quême, Roland. "A computer algorithm for finding new euclidean number fields." Journal de théorie des nombres de Bordeaux 10.1 (1998): 33-48. <http://eudml.org/doc/248166>.

@article{Quême1998,
abstract = {This article describes a computer algorithm which exhibits a sufficient condition for a number field to be euclidean for the norm. In the survey [3] p 405, Franz Lemmermeyer pointed out that 743 number fields where known (march 1994) to be euclidean (the first one, $\mathbb \{Q\}$, discovered by Euclid, three centuries B.C.!). In the first months of 1997, we found more than 1200 new euclidean number fields of degree 4, 5 and 6 with a computer algorithm involving classical lattice properties of the embedding of the degree $n$ field $\mathbf \{K\}$ into $\mathbb \{R\}^n$ and the structure of the unit group of $\mathbf \{K\}$. This articles ends with a generalization of the method for the determination of rings of $S$-integers of number fields euclidean for the norm and for the study of the inhomogeneous minimum of the norm form. Our results are in accordance with known results.},
author = {Quême, Roland},
journal = {Journal de théorie des nombres de Bordeaux},
keywords = {Euclidean algorithm; Euclidean number fields; -integers; inhomogeneous minimum},
language = {eng},
number = {1},
pages = {33-48},
publisher = {Université Bordeaux I},
title = {A computer algorithm for finding new euclidean number fields},
url = {http://eudml.org/doc/248166},
volume = {10},
year = {1998},
}

TY - JOUR
AU - Quême, Roland
TI - A computer algorithm for finding new euclidean number fields
JO - Journal de théorie des nombres de Bordeaux
PY - 1998
PB - Université Bordeaux I
VL - 10
IS - 1
SP - 33
EP - 48
AB - This article describes a computer algorithm which exhibits a sufficient condition for a number field to be euclidean for the norm. In the survey [3] p 405, Franz Lemmermeyer pointed out that 743 number fields where known (march 1994) to be euclidean (the first one, $\mathbb {Q}$, discovered by Euclid, three centuries B.C.!). In the first months of 1997, we found more than 1200 new euclidean number fields of degree 4, 5 and 6 with a computer algorithm involving classical lattice properties of the embedding of the degree $n$ field $\mathbf {K}$ into $\mathbb {R}^n$ and the structure of the unit group of $\mathbf {K}$. This articles ends with a generalization of the method for the determination of rings of $S$-integers of number fields euclidean for the norm and for the study of the inhomogeneous minimum of the norm form. Our results are in accordance with known results.
LA - eng
KW - Euclidean algorithm; Euclidean number fields; -integers; inhomogeneous minimum
UR - http://eudml.org/doc/248166
ER -

References

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  1. [1] S. Cavallar and F. Lemmermeyer, The euclidean algorithm in cubic number fields, draft (August 1996). Zbl0918.11057
  2. [2] H. Davenport, Linear forms associated with an algebraic number field, Quarterly J. Math., (2) 3, (1952), pp. 32-41. Zbl0047.27402MR47707
  3. [3] F. Lemmermeyer, The euclidean algorithm in algebraic number fields, Expo. Mat., no 13 (1995), pp. 385-416. Zbl0843.11046MR1362867
  4. [4] H.W. Lenstra, Euclidean number fields of large degree, Invent. Math., n0 38, (1977), pp. 237-254. Zbl0328.12007MR429826
  5. [5] O.T. O'Meara, On the finite generation of linear groups over Hasse domains, J. Reine Angew. Math. n0 217 (1965), pp. 79-108. Zbl0128.25502MR179265
  6. [6] P. Samuel, Theorie algébrique des nombres, Hermann, (1967). Zbl0146.06402MR215808

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