Approximatting rings of integers in number fields

J. A. Buchmann; H. W. Lenstra

Journal de théorie des nombres de Bordeaux (1994)

  • Volume: 6, Issue: 2, page 221-260
  • ISSN: 1246-7405

Abstract

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In this paper we study the algorithmic problem of finding the ring of integers of a given algebraic number field. In practice, this problem is often considered to be well-solved, but theoretical results indicate that it is intractable for number fields that are defined by equations with very large coefficients. Such fields occur in the number field sieve algorithm for factoring integers. Applying a variant of a standard algorithm for finding rings of integers, one finds a subring of the number field that one may view as the “best guess” one has for the ring of integers. This best guess is probably often correct. Our main concern is what can be proved about this subring. We show that it has a particularly transparent local structure, which is reminiscent of the structure of tamely ramified extensions of local fields. A major portion of the paper is devoted to the study of rings that are “tame” in our more general sense. As a byproduct, we prove complexity results that elaborate upon a result of Chistov. The paper also includes a section that discusses polynomial time algorithms related to finitely generated abelian groups.

How to cite

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Buchmann, J. A., and Lenstra, H. W.. "Approximatting rings of integers in number fields." Journal de théorie des nombres de Bordeaux 6.2 (1994): 221-260. <http://eudml.org/doc/247529>.

@article{Buchmann1994,
abstract = {In this paper we study the algorithmic problem of finding the ring of integers of a given algebraic number field. In practice, this problem is often considered to be well-solved, but theoretical results indicate that it is intractable for number fields that are defined by equations with very large coefficients. Such fields occur in the number field sieve algorithm for factoring integers. Applying a variant of a standard algorithm for finding rings of integers, one finds a subring of the number field that one may view as the “best guess” one has for the ring of integers. This best guess is probably often correct. Our main concern is what can be proved about this subring. We show that it has a particularly transparent local structure, which is reminiscent of the structure of tamely ramified extensions of local fields. A major portion of the paper is devoted to the study of rings that are “tame” in our more general sense. As a byproduct, we prove complexity results that elaborate upon a result of Chistov. The paper also includes a section that discusses polynomial time algorithms related to finitely generated abelian groups.},
author = {Buchmann, J. A., Lenstra, H. W.},
journal = {Journal de théorie des nombres de Bordeaux},
keywords = {maximal order; tame extensions; algorithm; computing the maximal order of an algebraic number field; survey; polynomial time algorithms},
language = {eng},
number = {2},
pages = {221-260},
publisher = {Université Bordeaux I},
title = {Approximatting rings of integers in number fields},
url = {http://eudml.org/doc/247529},
volume = {6},
year = {1994},
}

TY - JOUR
AU - Buchmann, J. A.
AU - Lenstra, H. W.
TI - Approximatting rings of integers in number fields
JO - Journal de théorie des nombres de Bordeaux
PY - 1994
PB - Université Bordeaux I
VL - 6
IS - 2
SP - 221
EP - 260
AB - In this paper we study the algorithmic problem of finding the ring of integers of a given algebraic number field. In practice, this problem is often considered to be well-solved, but theoretical results indicate that it is intractable for number fields that are defined by equations with very large coefficients. Such fields occur in the number field sieve algorithm for factoring integers. Applying a variant of a standard algorithm for finding rings of integers, one finds a subring of the number field that one may view as the “best guess” one has for the ring of integers. This best guess is probably often correct. Our main concern is what can be proved about this subring. We show that it has a particularly transparent local structure, which is reminiscent of the structure of tamely ramified extensions of local fields. A major portion of the paper is devoted to the study of rings that are “tame” in our more general sense. As a byproduct, we prove complexity results that elaborate upon a result of Chistov. The paper also includes a section that discusses polynomial time algorithms related to finitely generated abelian groups.
LA - eng
KW - maximal order; tame extensions; algorithm; computing the maximal order of an algebraic number field; survey; polynomial time algorithms
UR - http://eudml.org/doc/247529
ER -

References

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