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On the calculation of the units of algebraic number fields

Published online by Cambridge University Press:  22 January 2016

Takashi Azuhata
Takashi Azuhata*
Affiliation:
Department of Mathematics Science, University of Tokyo, 26 Wakamiya, Shinjuku-ku Tokyo, Japan
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The method to find a system of fundamental units were given by G. F. Voronoi in the case of purely cubic fields (see B. N. Delone and D. K. Faddeev [1]), and by K. K. Biilebic [2] in general case of algebraic number fields except for quadratic fields. But they are rather complicated for direct calculation. On the other hand, in some special cases, units can be calculated by the Jacobi-Perron algorithm which is a generalization of the continued fractional expansion (see, for example, L. Bernstein [3]). In this paper, we will show the new method to calculate a system of independent units by multiplying several numbers. We will give some examples for purely cubic fields in Section 3.

Type
Research Article
Information
Nagoya Mathematical Journal , Volume 101 , March 1986 , pp. 181 - 185
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 1986

References

[ 1 ] Delone, B. N. and Faddeev, D.K., The theory of irrationalities of the third degree, Trudy Mat. Inst. Steklov, (1940), Trans. Amer. Math. Soc, 10 (1955).Google Scholar
[ 2 ] Billebic, K. K., On the units of algebraic fields of third and fourth degree, Mat. Sb., (1956) 123–136.Google Scholar
[ 3 ] Bernstein, L., The Jacobi-Perron algorithm, its theory and application, Lecture Note in Math., Springer, 207 (1971).Google Scholar