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Voronoi’s algorithm in purely cubic congruence function fields of unit rank 1. (English) Zbl 1042.11068
Summary: The first part of this paper classifies all purely cubic function fields over a finite field of characteristic not equal to 3. In the remainder, we describe a method for computing the fundamental unit and regulator of a purely cubic congruence function field of unit rank 1 and characteristic at least 5. The technique is based on Voronoi’s algorithm for generating a chain of successive minima in a multiplicative cubic lattice, which is used for calculating the fundamental unit and regulator of a purely cubic number field.

MSC:
11R16 Cubic and quartic extensions
11R27 Units and factorization
11R58 Arithmetic theory of algebraic function fields
11-04 Software, source code, etc. for problems pertaining to number theory
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[1] J. A. Buchmann, A generalization of Voronoi’s algorithm I, II. J. Number Theory 20 (1985), 177-209. · Zbl 0575.12005
[2] Johannes Buchmann, The computation of the fundamental unit of totally complex quartic orders, Math. Comp. 48 (1987), no. 177, 39 – 54. · Zbl 0627.12004
[3] Johannes Buchmann, On the computation of units and class numbers by a generalization of Lagrange’s algorithm, J. Number Theory 26 (1987), no. 1, 8 – 30. · Zbl 0615.12001 · doi:10.1016/0022-314X(87)90092-8 · doi.org
[4] Johannes Buchmann, On the period length of the generalized Lagrange algorithm, J. Number Theory 26 (1987), no. 1, 31 – 37. · Zbl 0615.12002 · doi:10.1016/0022-314X(87)90093-X · doi.org
[5] J. A. Buchmann, Zur Komplexität der Berechnung von Einheiten und Klassenzahlen algebraischer Zahlkörper. Habilitationsschrift, Universität Düsseldorf, Germany 1987.
[6] Johannes Buchmann and H. C. Williams, On the infrastructure of the principal ideal class of an algebraic number field of unit rank one, Math. Comp. 50 (1988), no. 182, 569 – 579. · Zbl 0653.12005
[7] B. N. Delone and D. K. Faddeev, The theory of irrationalities of the third degree, Translations of Mathematical Monographs, Vol. 10, American Mathematical Society, Providence, R.I., 1964. · Zbl 0133.30202
[8] Max Deuring, Lectures on the theory of algebraic functions of one variable, Lecture Notes in Mathematics, Vol. 314, Springer-Verlag, Berlin-New York, 1973. · Zbl 0249.14008
[9] E. Jung, Theorie der Algebraischen Funktionen einer Veränderlichen. Berlin 1923. · JFM 49.0261.04
[10] M. Mang, Berechnung von Fundamentaleinheiten in algebraischen, insbesondere rein-kubischen Kongruenzfunktionenkörpern. Diplomarbeit, Universität des Saarlandes, Saarbrücken, Germany 1987.
[11] M. Pohst and H. Zassenhaus, Algorithmic algebraic number theory, Encyclopedia of Mathematics and its Applications, vol. 30, Cambridge University Press, Cambridge, 1997. Revised reprint of the 1989 original. · Zbl 0685.12001
[12] F. K. Schmidt, Analytische Zahlentheorie in Körpern der Charakteristik \(p\). Math. Zeitschrift 33 (1931), 1-32. · JFM 57.0229.02
[13] Daniel Shanks, The infrastructure of a real quadratic field and its applications, Proceedings of the Number Theory Conference (Univ. Colorado, Boulder, Colo., 1972) Univ. Colorado, Boulder, Colo., 1972, pp. 217 – 224. · Zbl 0334.12005
[14] A. Stein, Baby Step-Giant Step-Verfahren in reell-quadratischen Kongruenzfunktionenkörpern mit Charakteristik ungleich 2. Diplomarbeit, Universität des Saarlandes, Saarbrücken, Germany 1992.
[15] A. Stein & H. C. Williams, Some methods for evaluating the regulator of a real quadratic function field. To appear in Exp. Math. · Zbl 0987.11071
[16] Henning Stichtenoth, Algebraic function fields and codes, Universitext, Springer-Verlag, Berlin, 1993. · Zbl 0816.14011
[17] G. F. Voronoi, On a Generalization of the Algorithm of Continued Fractions (in Russian). Doctoral Dissertation, Warsaw 1896.
[18] Bosco Weis and Horst G. Zimmer, Artins Theorie der quadratischen Kongruenzfunktionenkörper und ihre Anwendung auf die Berechnung der Einheiten- und Klassengruppen, Mitt. Math. Ges. Hamburg 12 (1991), no. 2, 261 – 286 (German). Mathematische Wissenschaften gestern und heute. 300 Jahre Mathematische Gesellschaft in Hamburg, Teil 2. · Zbl 0757.11046
[19] Hugh C. Williams, Continued fractions and number-theoretic computations, Rocky Mountain J. Math. 15 (1985), no. 2, 621 – 655. Number theory (Winnipeg, Man., 1983). · Zbl 0594.12003 · doi:10.1216/RMJ-1985-15-2-621 · doi.org
[20] H. C. Williams, G. Cormack, and E. Seah, Calculation of the regulator of a pure cubic field, Math. Comp. 34 (1980), no. 150, 567 – 611. · Zbl 0431.12006
[21] H. C. Williams, G. W. Dueck, and B. K. Schmid, A rapid method of evaluating the regulator and class number of a pure cubic field, Math. Comp. 41 (1983), no. 163, 235 – 286. · Zbl 0528.12004
[22] H. C. Williams and M. C. Wunderlich, On the parallel generation of the residues for the continued fraction factoring algorithm, Math. Comp. 48 (1987), no. 177, 405 – 423. · Zbl 0617.10005
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