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Mathematics of Computation

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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 $ Q(\sqrt[3]{D})$. 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 $ D\;( \leqslant {10^5})$ such that the class number of $ Q\;(\sqrt[3]{D})$ is not divisible by 3. Several tables summarizing the results of this computation are also presented.

References [Enhancements On Off] (What's this?)

  • [1] PIERRE BARRUCAND, H. C. WILLIAMS & L. BANIUK, ``A computational technique for determining the class number of a pure cubic field,'' Math. Comp., v. 30, 1976, pp. 312-323. MR 0392913 (52:13726)
  • [2] B. D. BEACH, H. C. WILLIAMS & C. R. ZARNKE, ``Some computer results on units in quadratic and cubic fields,'' Proc. Twenty-Fifth Summer Meeting of the Canadian Math. Congress, Lakehead Univ., Thunder Bay, Ont., 1971, pp. 609-648. MR 0337887 (49:2656)
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Article copyright: © Copyright 1980 American Mathematical Society

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