Determination of principal factors in and

Author:
H. C. Williams

Journal:
Math. Comp. **38** (1982), 261-274

MSC:
Primary 12A30; Secondary 12A45

MathSciNet review:
637306

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Abstract | References | Similar Articles | Additional Information

Abstract: Let 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 . 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.

**[1]**Pierre Barrucand and Harvey Cohn,*A rational genus, class number divisibility, and unit theory for pure cubic fields*, J. Number Theory**2**(1970), 7–21. MR**0249398****[2]**Pierre Barrucand and Harvey Cohn,*Remarks on principal factors in a relative cubic field*, J. Number Theory**3**(1971), 226–239. MR**0276197****[3]**Horst Brunotte, Jutta Klingen, and Manfred Steurich,*Einige Bemerkungen zu Einheiten in reinen kubischen Körpern*, Arch. Math. (Basel)**29**(1977), no. 2, 154–157 (German). MR**0457399****[4]**B. N. Delone and D. K. Faddeev,*The theory of irrationalities of the third degree*, Translations of Mathematical Monographs, Vol. 10, American Mathematical Society, Providence, R.I., 1964. MR**0160744****[5]**Franz Halter-Koch,*Eine Bemerkung über kubische Einheiten*, Arch. Math. (Basel)**27**(1976), no. 6, 593–595. MR**0429827****[6]**Patrick Morton,*On Rédei’s theory of the Pell equation*, J. Reine Angew. Math.**307/308**(1979), 373–398. MR**534233**, 10.1515/crll.1979.307-308.373**[7]**J. C. Lagarias,*On the computational complexity of determining the solvability or unsolvability of the equation 𝑋²-𝐷𝑌²=-1*, Trans. Amer. Math. Soc.**260**(1980), no. 2, 485–508. MR**574794**, 10.1090/S0002-9947-1980-0574794-0**[8]**G. F. Voronoi,*On a Generalization of the Algorithm of Continued Fractions*, Doctoral Dissertation, Warsaw, 1896. (Russian)**[9]**H. C. Williams and J. Broere,*A computational technique for evaluating 𝐿(1,𝜒) and the class number of a real quadratic field*, Math. Comp.**30**(1976), no. 136, 887–893. MR**0414522**, 10.1090/S0025-5718-1976-0414522-5**[10]**H. C. Williams, G. Cormack, and E. Seah,*Calculation of the regulator of a pure cubic field*, Math. Comp.**34**(1980), no. 150, 567–611. MR**559205**, 10.1090/S0025-5718-1980-0559205-7**[11]**H. C. Williams,*Improving the speed of calculating the regulator of certain pure cubic fields*, Math. Comp.**35**(1980), no. 152, 1423–1434. MR**583520**, 10.1090/S0025-5718-1980-0583520-4**[12]**H. C. Williams,*Some results concerning Voronoĭ’s continued fraction over 𝑄(\root3\of{𝐷})*, Math. Comp.**36**(1981), no. 154, 631–652. MR**606521**, 10.1090/S0025-5718-1981-0606521-7

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DOI:
http://dx.doi.org/10.1090/S0025-5718-1982-0637306-4

Keywords:
Principal factors,
Voronoi's algorithm,
Diophantine equations

Article copyright:
© Copyright 1982
American Mathematical Society