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

Abstract

Let l = 2 l = 2 or 3 and let D be a positive l-power-free integer. Also, let R be the product of all the rational primes which completely ramify in K = Q ( D 1 / l ) K = \mathcal {Q}({D^{1/l}}) . The integer d is a principal factor of the discriminant of K if d = N ( α ) d = N(\alpha ) , where α \alpha is an algebraic integer of K and d | R l − 1 d|{R^{l - 1}} . In this paper algorithms for finding these principal factors are described. Special attention is given to the case of l = 3 l = 3 , where it is shown that Voronoi’s continued fraction algorithm can be used to find principal factors. Some results of a computer search for principal factors for all Q ( D 3 ) \mathcal {Q}(\sqrt [3]{D}) with 2 ⩽ D ⩽ 15000 2 \leqslant D \leqslant 15000 are also presented.