# A computer algorithm for finding new euclidean number fields

Journal de théorie des nombres de Bordeaux (1998)

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

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topQuême, 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

top- [1] S. Cavallar and F. Lemmermeyer, The euclidean algorithm in cubic number fields, draft (August 1996). Zbl0918.11057
- [2] H. Davenport, Linear forms associated with an algebraic number field, Quarterly J. Math., (2) 3, (1952), pp. 32-41. Zbl0047.27402MR47707
- [3] F. Lemmermeyer, The euclidean algorithm in algebraic number fields, Expo. Mat., no 13 (1995), pp. 385-416. Zbl0843.11046MR1362867
- [4] H.W. Lenstra, Euclidean number fields of large degree, Invent. Math., n0 38, (1977), pp. 237-254. Zbl0328.12007MR429826
- [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] P. Samuel, Theorie algébrique des nombres, Hermann, (1967). Zbl0146.06402MR215808

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