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

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 \iD. This 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 D = 200171999, it required only 5 minutes for an AMDAHL 470/ VI computer to find that R = 518594546.969083 and h = 1. This same calculation would require about 8 days of computer time if it used only the standard Voronoi algorithm. a>0, and gcd(w,, m2, m3, «,, n2, n3,a) = 1. Also, if e = m2n3 — m3n2, then e | a. If u E %, we denote its conjugates by w' and «". We also write the norm of « as Nia) and define it as Nico) = »oco'w". From the simple fact that