# Approximatting rings of integers in number fields

Journal de théorie des nombres de Bordeaux (1994)

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

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topBuchmann, 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 -

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