Ideal arithmetic and infrastructure in purely cubic function fields
Journal de théorie des nombres de Bordeaux, Tome 13 (2001) no. 2, p. 609-631
Dans cet article, nous étudions l’arithmétique des idéaux fractionnnaires dans les corps de fonctions cubiques purs, ainsi que l’infrastructure de la classe des idéaux principaux lorsque le groupe des unités du corps est de rang 1. Nous décrivons d’abord la décomposition des polynômes irréductibles dans l’ordre maximal du corps. Nous construisons ensuite des bases d’idéaux, dites canoniques, bien adaptées pour les calcul. Nous énonçons des algorithmes permettant de multiplier les idéaux, et même de les réduire lorsque le groupe des unités est de rang 1 et la caractéristique au moins 5, L’article se termine avec une analyse de l’infrastructure de l’ensemble des idéaux fractionnaires réduits principaux dans le cas des corps cubiques purs de groupe des unités de rang 1 et de caractéristique au moins 5.
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.
@article{JTNB_2001__13_2_609_0,
     author = {Scheidler, Renate},
     title = {Ideal arithmetic and infrastructure in purely cubic function fields},
     journal = {Journal de th\'eorie des nombres de Bordeaux},
     publisher = {Universit\'e Bordeaux I},
     volume = {13},
     number = {2},
     year = {2001},
     pages = {609-631},
     zbl = {0995.11064},
     mrnumber = {1879675},
     language = {en},
     url = {http://http://www.numdam.org/item/JTNB_2001__13_2_609_0}
}
Scheidler, Renate. Ideal arithmetic and infrastructure in purely cubic function fields. Journal de théorie des nombres de Bordeaux, Tome 13 (2001) no. 2, pp. 609-631. http://www.numdam.org/item/JTNB_2001__13_2_609_0/

[1] M. Bauer, The arithmetic of certain cubic function fields. Submitted to Math. Comp. | MR 2034129 | Zbl 1053.11087

[2] B.N. Delone, D.K. Faddeev, The theory of irrationalities of the third degree. Transl. Math. Monographs 10, Amer. Math. Soc., Providence (Rhode Island), 1964. | MR 160744 | Zbl 0133.30202

[3] D. Shanks, The infrastructure of a real quadratic field and its applications. Proc. 1972 Number Theory Conf., Boulder (Colorado) 1972, 217-224. | MR 389842 | Zbl 0334.12005

[4] R. Scheidler, Reduction in purely cubic function fields of unit rank one. Proc. Fourth Algorithmic Number Theory Symp. ANTS-IV, Lect. Notes Comp. Science 1838, Springer, Berlin, 2000, 151-532. | MR 1850630 | Zbl 1035.11057

[5] R. Scheidler, A. Stein, Voronoi's algorithm in purely cubic congruence function fields of unit rank 1. Math. Comp. 69 (2000), 1245-1266. | MR 1653974 | Zbl 1042.11068

[6] A. Stein, H.C. Williams, Some methods for evaluating the regulator of a real quadratic function field. Exp. Math. 8 (1999), 119-133. | MR 1700574 | Zbl 0987.11071

[7] H. Stichtenoth, Algebraic function fields and codes. Universitext, Springer-Verlag, Berlin, 1993. | MR 1251961 | Zbl 0816.14011

[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. | MR 823273 | Zbl 0594.12003

[11] H.C. Williams, G. Cormack, E. Seah, Calculation of the regulator of a pure cubic field. Math. Comp. 34 (1980), 567-611. | MR 559205 | Zbl 0431.12006

[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. | MR 701638 | Zbl 0528.12004