Calculation of the regulator of a pure cubic field

Authors:
H. C. Williams, G. Cormack and E. Seah

Journal:
Math. Comp. **34** (1980), 567-611

MSC:
Primary 12A45; Secondary 12A30, 12A50

MathSciNet review:
559205

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Abstract: A description is given of a modified version of Voronoi's algorithm for obtaining the regulator of a pure cubic field . This new algorithm has the advantage of executing relatively rapidly for large values of *D*. It also eliminates a computational problem which occurs in almost all algorithms for finding units in algebraic number fields. This is the problem of performing calculations involving algebraic irrationals by using only approximations of these numbers.

The algorithm was implemented on a computer and run on all values of such that the class number of is not divisible by 3. Several tables summarizing the results of this computation are also presented.

**[1]**Pierre Barrucand, H. C. Williams, and L. Baniuk,*A computational technique for determining the class number of a pure cubic field*, Math. Comp.**30**(1976), no. 134, 312–323. MR**0392913**, 10.1090/S0025-5718-1976-0392913-9**[2]**B. D. Beach, H. C. Williams, and C. R. Zarnke,*Some computer results on units in quadratic and cubic fields*, Proceedings of the Twenty-Fifth Summer Meeting of the Canadian Mathematical Congress (Lakehead Univ., Thunder Bay, Ont., 1971) Lakehead Univ., Thunder Bay, Ont., 1971, pp. 609–648. MR**0337887****[3]**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****[4]**Taira Honda,*Pure cubic fields whose class numbers are multiples of three*, J. Number Theory**3**(1971), 7–12. MR**0292795****[5]**Ray Steiner,*On the units in algebraic number fields*, Proceedings of the Sixth Manitoba Conference on Numerical Mathematics (Univ. Manitoba, Winnipeg, Man., 1976) Congress. Numer., XVIII, Utilitas Math., Winnipeg, Man., 1977, pp. 413–435. MR**532716****[6]**G. F. VORONOI,*On a Generalization of the Algorithm of Continued Fractions*, Doctoral Dissertation, Warsaw, 1896. (Russian)**[7]**Hideo Wada,*A table of fundamental units of purely cubic fields*, Proc. Japan Acad.**46**(1970), 1135–1140. MR**0294292****[8]**H. C. Williams,*Certain pure cubic fields with class-number one*, Math. Comp.**31**(1977), no. 138, 578–580. MR**0432591**, 10.1090/S0025-5718-1977-0432591-4**[9]**H. C. Williams and Daniel Shanks,*A note on class-number one in pure cubic fields*, Math. Comp.**33**(1979), no. 148, 1317–1320. MR**537977**, 10.1090/S0025-5718-1979-0537977-7

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DOI:
http://dx.doi.org/10.1090/S0025-5718-1980-0559205-7

Article copyright:
© Copyright 1980
American Mathematical Society