Determination of principal factors in and
Author:
H. C. Williams
Journal:
Math. Comp. 38 (1982), 261274
MSC:
Primary 12A30; Secondary 12A45
MathSciNet review:
637306
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Abstract 
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Abstract: Let or 3 and let D be a positive lpowerfree integer. Also, let R be the product of all the rational primes which completely ramify in . The integer d is a principal factor of the discriminant of K if , where is an algebraic integer of K and . In this paper algorithms for finding these principal factors are described. Special attention is given to the case of , 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 with are also presented.
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 Pierre Barrucand & Harvey Cohn, ``A rational genus, class divisibility, and unit theory for pure cubic fields,'' J. Number Theory, v. 2, 1970, pp. 721. MR 0249398 (40:2643)
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 Pierre Barrucand & Harvey Cohn, ``Remarks on principal factors in a relative cubic field,'' J. Number Theory, v. 3, 1971, pp. 226239. MR 0276197 (43:1945)
 [3]
 H. Brunotte, J. Klingen & M. Steurich, ``Einige Bemerkungen zu Einheiten in reinen kubischen Körpern,'' Arch. Math., v. 29, 1977, pp. 154157. MR 0457399 (56:15604)
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 J. C. Lagarias, ``On the computational complexity of determining the solvability of the equation ,'' Trans. Amer. Math. Soc., v. 260, 1980, pp. 485508. MR 574794 (81g:10029)
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 G. F. Voronoi, On a Generalization of the Algorithm of Continued Fractions, Doctoral Dissertation, Warsaw, 1896. (Russian)
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 H. C. Williams & J. Broere, ``A computational technique for evaluating and the class number of a real quadratic field,'' Math. Comp., v. 30, 1976, pp. 887893. MR 0414522 (54:2623)
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 H. C. Williams, G. Cormack & E. Seah, ``Computation of the regulator of a pure cubic field,'' Math. Comp., v. 34, 1980, pp. 567611. MR 559205 (81d:12003)
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 H. C. Williams, ``Improving the speed of calculating the regulator of certain pure cubic fields,'' Math. Comp., v. 35, 1980, pp. 14231434. MR 583520 (82a:12003)
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 H. C. Williams, ``Some results concerning Voronoi's continued fraction over ,'' Math. Comp., v. 36, 1981, pp. 631652. MR 606521 (82j:12011)
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00255718198206373064
PII:
S 00255718(1982)06373064
Keywords:
Principal factors,
Voronoi's algorithm,
Diophantine equations
Article copyright:
© Copyright 1982
American Mathematical Society
