Calculation of the Regulator of a Pure Cubic Field

Abstract

JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact support@jstor.org.. American Mathematical Society is collaborating with JSTOR to digitize, preserve and extend access to Mathematics of Computation.Abstract. A description is given of a modified version of Voronoi's algorithm for obtaining the regulator of a pure cubic field Q( 3D). 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 implemented on a computer and run on all values of D (< 105) such that the class number of Q(./D) is not divisible by 3. Several tables summarizing the results of this computation are also presented.

A note on class-number one in pure cubic fields

“…This is reported elsewhere [5], together with other tables computed with its use. With the faster algorithm, the smaller discriminants in the K = Qi\/r), and their more restricted population, we have now computed these fields for all r < 200,000.…”

Determination of principal factors in 𝒬(√𝒟) and 𝒬(\root3\of𝒟)

“…This method is described in Delone and Faddeev [4]. In Williams, Cormack, and Seah [10] a proof is provided of a relatively rapid technique for finding these 0gw's.…”

“…Using the results of [2] together with a later result of Halter-Koch [5], we get the following Theorem. Consider the equation (1)(2)(3)(4)(5)(6)(7)(8)(9)(10)(11)(12)(13)(14) NL/a(E) = p.…”

Ideal arithmetic and infrastructure in purely cubic function fields

“…This is certainly the case for purely cubic function field representations of unit rank one, the function field analogue of purely cubic number fields. Ideal arithmetic and the infrastructure in purely cubic number fields were investigated by Williams et al in [11,12,10], and much of the work in this paper was guided by these sources. As in the case of quadratic function fields, the infrastructure can be used to compute the regulator and the ideal class number, and hence the order of the group of rational points of the Jacobian of a purely cubic function field.…”