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Computation of the fundamental units and the regulator of a cyclic cubic function field. (English) Zbl 1064.11082

The authors present algorithms based on Voronoi’s method for computing the fundamental units and the regulator of cyclic cubic function fields over a finite field of order \(q \equiv 1\bmod 3\). The paper also contains tables with the results of their calculations. For example, the regulators of about 50 fields with \(7\leq q\leq 911\) are listed.

MSC:

11Y40 Algebraic number theory computations
11R58 Arithmetic theory of algebraic function fields
11R27 Units and factorization
11R16 Cubic and quartic extensions
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References:

[1] DOI: 10.1112/plms/s2-34.1.360 · Zbl 0005.34203 · doi:10.1112/plms/s2-34.1.360
[2] Bauer M., ”The Arithmetic of Certain Cubic Function Fields.” · Zbl 1053.11087
[3] DOI: 10.1016/0022-314X(85)90039-3 · doi:10.1016/0022-314X(85)90039-3
[4] Delone B. N., The Theory of Irrationalities of the Third Degree. Transl. Math. Monographs 10 (1964) · Zbl 0133.30202
[5] Mang M., ”Berechnung von Fundamental einheiten in algebraischen, insbesondere rein-kubischen Kongruenzfunktionenkörpern.” (1987)
[6] Pohst M., Algorithmic Algebraic Number Theory,, 1. ed. (1997) · Zbl 0685.12001
[7] Scheidler, R.”Reduction in Purely Cubic Function Fields of Unit Rank One.”515–532. 2000Berlin: Springer-Verlag. [Scheidler 00], In Proc. Fourth Algorithmic Number Theory Symp. ANTS-IV, Lect. Notes Comp. Sci. 1838 · Zbl 1035.11057
[8] DOI: 10.1090/S0025-5718-99-01136-9 · Zbl 1042.11068 · doi:10.1090/S0025-5718-99-01136-9
[9] Stein A., Experim. Math. 8 pp 119– (1999)
[10] Stichtenoth H., Algebraic Function Fields and Codes. (1993) · Zbl 0816.14011
[11] Voronoi G. F., Doctoral dissertation, in: ”On a Generalization of the Algorithm of Continued Fractions (in Russian).” (1896)
[12] Williams, H. C. and Zarnke, C. R. ”Computer Calculation of Units in Cubic Fields.”. Proc. Second Manitoba Conf. Num. Math. Congressus Numerantium. Vol. VII, pp.433–468. Winnipeg, Manitoba: Utilitas Mathematica Publishing, Inc. [Williams et al. 73] · Zbl 0312.12012
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