Determining the fundamental unit of a pure cubic field given any unit
N. S. Jeans and M. D. Hendy
Math. Comp. 32 (1978), 925-935
Primary 12A30; Secondary 12A45
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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.
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
Sved, Units in pure cubic number fields, Ann. Univ. Sci.
Budapest. Eötvös Sect. Math. 13 (1970),
141–149 (1971). MR 0313199
Szekeres, Multidimensional continued fractions, Ann. Univ.
Sci. Budapest. Eötvös Sect. Math. 13 (1970),
113–140 (1971). MR 0313198
- B. D. BEACH, H. C. WILLIAMS & C. R. ZARNKE, "Some computer results on units in quadratic and cubic fields," Proc. 25th Summer Meeting Canad. Math. Congress, 1971, pp. 609-648. MR 0337887 (49:2656)
- MARTA SVED, "Units in pure cubic number fields," Ann. Univ. Sci. Budapest. Eötvös Sect. Math., v. 13, 1970, pp. 141-149. MR 0313199 (47:1754)
- G. SZEKERES, "Multidimensional continued fractions," Ann. Univ. Sci. Budapest. Eötvös Sect. Math., v. 13, 1970, pp. 113-140. MR 0313198 (47:1753)
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