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Parametrized units in algebraic number fields

Published online by Cambridge University Press:  24 October 2008

Loren D. Olson
Loren D. Olson
Affiliation:
Department of Mathematics, University of Tromsø, Norway
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Extract

One of the fundamental problems in algebraic number theory is the construction of units in algebraic number fields. Various authors have considered number fields which are parametrized by an integer variable. They have described units in these fields by polynomial expressions in the variable e.g. the fields ℚ(√[N2 + 1]), Nεℤ, with the units εN = N + √[N2 + l]. We begin this article by formulating a general principle for obtaining units in algebraic function fields and candidates for units in parametrized families of algebraic number fields. We show that many of the cases considered previously in the literature by such authors as Bernstein [2], Neubrand [8], and Stender [ll] fall in under this principle. Often the results may be obtained much more easily than before. We then examine the connection between parametrized cubic fields and elliptic curves. In §4 we consider parametrized quadratic fields, a situation previously studied by Neubrand [8]. We conclude in §5 by examining the effect of parametrizing the torsion structure on an elliptic curve at the same time.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 103 , Issue 1 , January 1988 , pp. 15 - 25
Copyright
Copyright © Cambridge Philosophical Society 1988

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References

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