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

Author: H. C. Williams
Journal: Math. Comp. 38 (1982), 261-274
MSC: Primary 12A30; Secondary 12A45
MathSciNet review: 637306
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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\vert{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.

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Keywords: Principal factors, Voronoi's algorithm, Diophantine equations
Article copyright: © Copyright 1982 American Mathematical Society