# Ideal arithmetic and infrastructure in purely cubic function fields

Journal de théorie des nombres de Bordeaux (2001)

- Volume: 13, Issue: 2, page 609-631
- ISSN: 1246-7405

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topScheidler, Renate. "Ideal arithmetic and infrastructure in purely cubic function fields." Journal de théorie des nombres de Bordeaux 13.2 (2001): 609-631. <http://eudml.org/doc/248728>.

@article{Scheidler2001,

abstract = {This paper investigates the arithmetic of fractional ideals of a purely cubic function field and the infrastructure of the principal ideal class when the field has unit rank one. First, we describe how irreducible polynomials decompose into prime ideals in the maximal order of the field. We go on to compute so-called canonical bases of ideals; such bases are very suitable for computation. We state algorithms for ideal multiplication and, in the case of unit rank one and characteristic at least five, ideal reduction. The paper concludes with an analysis of the infrastructure in the set of reduced fractional principal ideals of a purely cubic function field of unit rank one and characteristic at least five.},

author = {Scheidler, Renate},

journal = {Journal de théorie des nombres de Bordeaux},

keywords = {infrastructure; fractional ideal; purely cubic function field},

language = {eng},

number = {2},

pages = {609-631},

publisher = {Université Bordeaux I},

title = {Ideal arithmetic and infrastructure in purely cubic function fields},

url = {http://eudml.org/doc/248728},

volume = {13},

year = {2001},

}

TY - JOUR

AU - Scheidler, Renate

TI - Ideal arithmetic and infrastructure in purely cubic function fields

JO - Journal de théorie des nombres de Bordeaux

PY - 2001

PB - Université Bordeaux I

VL - 13

IS - 2

SP - 609

EP - 631

AB - This paper investigates the arithmetic of fractional ideals of a purely cubic function field and the infrastructure of the principal ideal class when the field has unit rank one. First, we describe how irreducible polynomials decompose into prime ideals in the maximal order of the field. We go on to compute so-called canonical bases of ideals; such bases are very suitable for computation. We state algorithms for ideal multiplication and, in the case of unit rank one and characteristic at least five, ideal reduction. The paper concludes with an analysis of the infrastructure in the set of reduced fractional principal ideals of a purely cubic function field of unit rank one and characteristic at least five.

LA - eng

KW - infrastructure; fractional ideal; purely cubic function field

UR - http://eudml.org/doc/248728

ER -

## References

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- [5] R. Scheidler, A. Stein, Voronoi's algorithm in purely cubic congruence function fields of unit rank 1. Math. Comp.69 (2000), 1245-1266. Zbl1042.11068MR1653974
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- [8] G.F. Voronoi, Concerning algebraic integers derivable from a root of an equation of the third degree (in Russian). Master's Thesis, St. Petersburg (Russia), 1894.
- [9] G.F. Voronoi, On a generalization of the algorithm of continued fractions (in Russian). Doctoral Dissertation, Warsaw (Poland), 1896.
- [10] H.C. Williams, Continued fractions and number-theoretic computations. Rocky Mountain J. Math.15 (1985), 621-655. Zbl0594.12003MR823273
- [11] H.C. Williams, G. Cormack, E. Seah, Calculation of the regulator of a pure cubic field. Math. Comp.34 (1980), 567-611. Zbl0431.12006MR559205
- [12] H.C. Williams, G.W. Dueck, B.K. Schmid, A rapid method of evaluating the regulator and class number of a pure cubic field. Math. Comp.41 (1983), 235-286. Zbl0528.12004MR701638

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