Abstract
This paper describes a method for computing the fundamental unit and regulator of a purely cubic congruence function field of unit rank 1. 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.
Research supported by NSF grant DMS-9631647
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Buchmann, J. A.: A generalization of Voronoi's algorithm I, II. J. Number Theory 20 (1985) 177–209
Buchmann, J. A.: The computation of the fundamental unit of totally complex quartic orders. Math. Comp. 48 (1987) 39–54
Buchmann, J. A.: On the computation of units and class numbers by a generalization of Lagrange's algorithm. J. Number Theory 26 (1987) 8–30
Buchmann, J. A.: On the period length of the generalized Lagrange algorithm. J. Number Theory 26 (1987) 31–37
Buchmann, J. A.: Zur KomplexitÄt der Berechnung von Einheiten und Klassenzahlen algebraischer Zahlkörper. Habilitationsschrift, UniversitÄt Düsseldorf, Germany, (1987)
Buchmann, J. A., Williams, H. C.: On the infrastructure of the principal ideal class of an algebraic number field of unit rank one. Math. Comp. 50 (1988) 569–579
Delone, B. N., Fadeev, D. K.: The Theory of Irrationalities of the Third Degree. Transi. Math. Monographs 10, Amer. Math. Soc., Providence, Rhode Island (1964)
Jung, E.: Theorie der Algebraischen Punktionen einer VerÄnderlichen. Berlin (1923)
Mang, M.: Berechnung von Fundamentaleinheiten in algebraischen, insbesondere rein-kubischen Kongruenzfunktionenkörpern. Diplomarbeit, UniversitÄt des Saarlandes, Saarbrücken, Germany, (1987)
Pohst, M., Zassenhaus, H.: Algorithmic Algebraic Number Theory. Cambridge University Press, 1st paperpack ed., Cambridge (1997)
Scheidler, R., Stein, A.: Voronoi's Algorithm in Purely Cubic Congruence Function Fields of Unit Rank 1 (in preparation)
Stein, A., Williams, H. C.: Some Methods for Evaluating the Regulator of a Real Quadratic Function Field. Experimental Mathematics (to appear)
Stichtenoth, H.: Algebraic Function Fields and Codes. Springer, Berlin (1993)
Voronoi, G. F.: On a Generalization of the Algorithm of Continued Fractions (in Russian). Doctoral Dissertation, Warsaw, Poland, (1896)
Williams, H. C: Continued fractions and number-theoretic computations. Rocky Mountain J. Math. 15 (1985) 621–655
Williams, H. C., Cormack, G., Seah, E.: Calculation of the regulator of a pure cubic field. Math. Comp. 34 (1980) 567–611
Williams, H. C., Dueck, G. W., Schmid, B. K.: A rapid method of evaluating the regulator and class number of a pure cubic field. Math. Comp. 41 (1983) 235–286
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© 1998 Springer-Verlag Berlin Heidelberg
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Scheidler, R., Stein, A. (1998). Unit computation in purely cubic function fields of unit rank 1. In: Buhler, J.P. (eds) Algorithmic Number Theory. ANTS 1998. Lecture Notes in Computer Science, vol 1423. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0054895
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DOI: https://doi.org/10.1007/BFb0054895
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