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Ideal arithmetic and infrastructure in purely cubic function fields. (English) Zbl 0995.11064
D. Shanks [Proc. 1972 Number Theory Conf., Univ. Colorado, Boulder 217-224 (1972; Zbl 0334.12005)] introduced the idea of an infrastructure with respect to the multiplication of reduced ideals in the ring of integers of a quadratic number field. The product of two reduced ideals is not necessarily reduced, but one can find a reduced ideal that is “very close to” the product. Shanks used this idea to give a faster algorithm for calculating the regulator of the number field. This idea was extended to the case of a pure cubic number field by H. C. Williams, G. W. Dueck and B. K. Schmid [Math. Comput. 41, 235-286 (1983; Zbl 0528.12004)]. The author carries out a similar program for purely cubic function fields of unit rank one and analyzes the infrastructure in the set of reduced fractional principal ideals of the maximal order in such a function field. Related results have also been obtained by Mark L. Bauer [The arithmetic of certain cubic function fields, Math. Comput. (to appear)].

MSC:
11R58 Arithmetic theory of algebraic function fields
14H52 Elliptic curves
13P99 Computational aspects and applications of commutative rings
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References:
[1] Bauer, M., The arithmetic of certain cubic function fields. Submitted to Math. Comp. · Zbl 1053.11087
[2] Delone, B.N., Faddeev, D.K., The theory of irrationalities of the third degree. Transl. Math. Monographs10, Amer. Math. Soc., Providence (Rhode Island), 1964. · Zbl 0133.30202
[3] Shanks, D., The infrastructure of a real quadratic field and its applications. Proc. 1972 Number Theory Conf., Boulder (Colorado)1972, 217-224. · Zbl 0334.12005
[4] Scheidler, R., Reduction in purely cubic function fields of unit rank one. Proc. Fourth Algorithmic Number Theory Symp. ANTS-IV, 1838, Springer, Berlin, 2000, 151-532. · Zbl 1035.11057
[5] Scheidler, R., Stein, A., Voronoi’s algorithm in purely cubic congruence function fields of unit rank 1. Math. Comp.69 (2000), 1245-1266. · Zbl 1042.11068
[6] Stein, A., Williams, H.C., Some methods for evaluating the regulator of a real quadratic function field. Exp. Math.8 (1999), 119-133. · Zbl 0987.11071
[7] Stichtenoth, H., Algebraic function fields and codes. Universitext, Springer-Verlag, Berlin, 1993. · Zbl 0816.14011
[8] Voronoi, G.F., Concerning algebraic integers derivable from a root of an equation of the third degree (in Russian). Master’s Thesis, St. Petersburg (Russia), 1894.
[9] Voronoi, G.F., On a generalization of the algorithm of continued fractions (in Russian). Doctoral Dissertation, Warsaw (Poland), 1896. · JFM 39.0274.01
[10] Williams, H.C., Continued fractions and number-theoretic computations. Rocky Mountain J. Math.15 (1985), 621-655. · Zbl 0594.12003
[11] Williams, H.C., Cormack, G., Seah, E., Calculation of the regulator of a pure cubic field. Math. Comp.34 (1980), 567-611. · Zbl 0431.12006
[12] 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. · Zbl 0528.12004
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