A Rapid Method of Evaluating the Regulator and Class Number of a Pure Cubic Field

H. C. Williams, G. W. Dueck, B. K. Schmid
1983 Mathematics of Computation  
Let % = 2(0) be the algebraic number field formed by adjoining 9 to the rationals 2. Let R and h be, respectively, the regulator and class number of %. Shanks has described a method of evaluating R for 2(vD ), where D is a positive integer. His technique improved the speed of the usual continued fraction algorithm for finding R by allowing one to proceed almost directly from the nth to the mth step, where m is approximately In, in the continued fraction expansion of \¡D . This paper shows how
more » ... s paper shows how Shanks' idea can be extended to the Voronoi algorithm, which is used to find R in cubic fields of negative discriminant. It also discusses at length an algorithm for finding R and h for pure cubic fields 1(fD), D an integer. Under a certain generalized Riemann Hypothesis the -ideas developed here will provide a new method which will find R and h in 0(02/5+e) operations. When h is small, this is an improvement over the 0(D/h) operations required by Voronoi's algorithm to find R. For example, with a 8"w" 8"2w"
doi:10.2307/2007780 fatcat:ntuowfkww5aztit7ge3cyaucwu