Determination of principal factors in $\mathcal {Q}(\sqrt {D})$ and $\mathcal {Q}(\root 3\of D)$

Abstract:

Let $l = 2$ 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 $K = \mathcal {Q}({D^{1/l}})$. The integer d is a principal factor of the discriminant of K if $d = N(\alpha )$, where $\alpha$ is an algebraic integer of K and $d|{R^{l - 1}}$. In this paper algorithms for finding these principal factors are described. Special attention is given to the case of $l = 3$, 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 $\mathcal {Q}(\sqrt [3]{D})$ with $2 \leqslant D \leqslant 15000$ are also presented.