Ideal arithmetic and infrastructure in purely cubic function fields

Renate Scheidler

Journal de théorie des nombres de Bordeaux (2001)

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

Abstract

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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.

How to cite

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Scheidler, 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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  1. [1] M. Bauer, The arithmetic of certain cubic function fields. Submitted to Math. Comp. Zbl1053.11087MR2034129
  2. [2] B.N. Delone, D.K. Faddeev, The theory of irrationalities of the third degree. Transl. Math. Monographs10, Amer. Math. Soc., Providence (Rhode Island), 1964. Zbl0133.30202MR160744
  3. [3] D. Shanks, The infrastructure of a real quadratic field and its applications. Proc. 1972 Number Theory Conf., Boulder (Colorado)1972, 217-224. Zbl0334.12005MR389842
  4. [4] R. Scheidler, Reduction in purely cubic function fields of unit rank one. Proc. Fourth Algorithmic Number Theory Symp. ANTS-IV, Lect. Notes Comp. Science1838, Springer, Berlin, 2000, 151-532. Zbl1035.11057MR1850630
  5. [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
  6. [6] A. Stein, H.C. Williams, Some methods for evaluating the regulator of a real quadratic function field. Exp. Math.8 (1999), 119-133. Zbl0987.11071MR1700574
  7. [7] H. Stichtenoth, Algebraic function fields and codes. Universitext, Springer-Verlag, Berlin, 1993. Zbl0816.14011MR1251961
  8. [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. [9] G.F. Voronoi, On a generalization of the algorithm of continued fractions (in Russian). Doctoral Dissertation, Warsaw (Poland), 1896. 
  10. [10] H.C. Williams, Continued fractions and number-theoretic computations. Rocky Mountain J. Math.15 (1985), 621-655. Zbl0594.12003MR823273
  11. [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. [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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