Determining the fundamental unit of a pure cubic field given any unit

Authors:
N. S. Jeans and M. D. Hendy

Journal:
Math. Comp. **32** (1978), 925-935

MSC:
Primary 12A30; Secondary 12A45

DOI:
https://doi.org/10.1090/S0025-5718-1978-0472761-3

MathSciNet review:
0472761

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Abstract: A number of algorithms which have been used to derive fundamental units for pure cubic fields suffer from the lack of absolute certainty that the units obtained are fundamental. We present here an algorithm which will correct this deficiency. Briefly, if is any nontrivial unit of a pure cubic field, then for some positive integer will be a fundamental unit. Our method determines which of the real numbers are integers of the field and, subsequently, will determine the coefficients of the fundamental unit. We illustrate the process with several numerical examples.

**[1]**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****[2]**Marta Sved,*Units in pure cubic number fields*, Ann. Univ. Sci. Budapest. Eötvös Sect. Math.**13**(1970), 141–149 (1971). MR**0313199****[3]**G. Szekeres,*Multidimensional continued fractions*, Ann. Univ. Sci. Budapest. Eötvös Sect. Math.**13**(1970), 113–140 (1971). MR**0313198**

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Additional Information

DOI:
https://doi.org/10.1090/S0025-5718-1978-0472761-3

Keywords:
Pure cubic fields,
fundamental unit,
Szekeres algorithm

Article copyright:
© Copyright 1978
American Mathematical Society