Calculation of the regulator of a pure cubic field

H. C. Williams, G. Cormack, E. Seah
1980 Mathematics of Computation
A description is given of a modified version of Voronoi's algorithm for obtaining the regulator of a pure cubic field (2(\A>)-This new algorithm has the advantage of executing relatively rapidly for large values of D. It also eliminates a computational problem which occurs in almost all algorithms for finding units in algebraic number fields. This is the problem of performing calculations involving algebraic irrationals by using only approximations of these numbers. The algorithm was
more » ... ithm was implemented on a computer and run on all values of D (< 10 ) such that the class number of Q_(\Jd) is not divisible by 3. Several tables summarizing the results of this computation are also presented. 1. Introduction. Let S be the real root of x3 -Bx2 +Cx-D = 0, an irreducible cubic equation with rational integer coefficients B, C, D and negative discriminant A. Let Q(6) be the cubic field formed by adjoining 5 to the rationals, and let Q{ §] be the ring of integers in Q(ô). The regulator of Q(5) is R = log e0, where e0 (>1) is the fundamental unit of Q(5). When B = C = 0, we say that Q(ô) is a pure cubic field. In Williams  an attempt was made to tabulate several pure cubic fields (%\$D) which have D a prime = -1 (mod 3) and class number 1. In doing this it was necessary to evaluate the value of R for each Q\$D). This was done by using the algorithm of Voronoi  as described in Delone and Faddeev [3, and Beach, Williams and Zarnke PL Calculations had to be terminated when D > 35100 for two reasons. The first reason was the immense amount of time needed (up to 10 minutes of C.P.U. time on an IBM 370-168 computer) to calculate an individual R value; the second, and more important reason, was that the values of D were getting too large for the precision available to the computer, even using double-precision arithmetic. All methods of evaluating R known to the authors, with one exception, require that it be possible to determine when an algebraic irrational a G Q(ô) exceeds zero. The computer can only calculate an approximation A to a, and when D is large it is not always true that HA >0, then a > 0.