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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)].

11R58 Arithmetic theory of algebraic function fields
14H52 Elliptic curves
13P99 Computational aspects and applications of commutative rings
Full Text: DOI EMIS Numdam EuDML
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