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A generalization of Voronoi’s unit algorithm. I, II. (English) Zbl 0575.12005
J. Number Theory 20, 177-191, 192-209 (1985).
By means of mapping an algebraic number field into Euclidean real space (parameter map) the author shows how to generalize the well-known unit algorithm of G. F. Voronoi [Doctoral dissertation (Warsaw 1896). Cf. also ch. 4 of B. N. Delone and D. K. Faddeev, The theory of irrationalities of the third degree (Transl. Math. Monogr. 10) (1964; Zbl 0133.302)] to all algebraic number fields. By the use of sequences of points called neighbours the author shows that the generalized Voronoi algorithm computes the fundamental units of all algebraic number fields of unit ranks 1 and 2. These two results represent an important step forward in the development of algorithms for finding units. The first is an extension of a result of R. J. Rudman and the reviewer [J. Number Theory 10, 16-34 (1978; Zbl 0372.12010)] who proved it only for quadratic and complex cubic fields, while the second extends Voronoi’s basic result for totally real cubic fields to all fields of unit rank 2.
The author also gives several tables of units of various fields - all units being computed by his algorithm. Unfortunately, he does not indicate in either of these two papers how one computes the neighbour of a given point. An inefficient method for this purpose was given by the reviewer [Numerical mathematics, Proc. 6th Manitoba Conf., Congr. Numerantium 18, 413-435 (1977; Zbl 0477.12003)].
Reviewer: R.Ph.Steiner

MSC:
11R27 Units and factorization
12-04 Software, source code, etc. for problems pertaining to field theory
References:
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