A rapid method of evaluating the regulator and class number of a pure cubic field

Abstract:

Let $\mathcal {K} = \mathcal {Q}(\theta )$ be the algebraic number field formed by adjoining $\theta$ to the rationals $\mathcal {Q}$. Let R and h be, respectively, the regulator and class number of $\mathcal {K}$. Shanks has described a method of evaluating R for $\mathcal {Q}(\sqrt D )$, 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 2n, in the continued fraction expansion of $\sqrt D$. 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 $\mathcal {Q}(\sqrt [3]{D})$, 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 $O({D^{2/5 + \varepsilon }})$ operations. When h is small, this is an improvement over the $O(D/h)$ operations required by Voronoi’s algorithm to find R. For example, with $D = 200171999$, it required only 5 minutes for an AMDAHL 470/V7 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.