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Unit computation in purely cubic function fields of unit rank 1. (English) Zbl 0935.11051
Buhler, J. P. (ed.), Algorithmic number theory. 3rd international symposium, ANTS-III, Portland, OR, USA, June 21-25, 1998. Proceedings. Berlin: Springer. Lect. Notes Comput. Sci. 1423, 592-606 (1998).
The authors generalize Voronoi’s algorithm for the computation of fundamental units in number fields with unit rank 1 [see J. Buchmann, J. Number Theory 20, 177-191, 192-209 (1985; Zbl 0575.12005)] to the case of pure cubic function fields of unit rank 1 (excluding the normal case and the characteristic 2 case).
After establishing the notion of a minimum of an ideal (adjacent to some element), they define reduced ideals. The definitions are basically the same as in the number field case. Using this, they give a complete algorithm for the computation of a fundamental unit. Additionally a modification is given to just compute the regulator.
The key ingredient of their algorithm is contained in section 4, where they give an explicit algorithm to compute a suitable reduced ideal basis from which the minimum can be obtained. Before giving some numerical examples, they discuss details important to an implementation.
For the entire collection see [Zbl 0891.00022].

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