On the existence ofp-units and minkowski units in totally real cyclic fields

  • F. Marko
Article

Abstract

LetK be a totally real cyclic number field of degree n > 1. A unit inK is called an m-unit, if the index of the group generated by its conjugations in the group U*K of all units modulo ±1 is coprime tom. It is proved thatK contains an m-unit for every m coprime to n.

The mutual relationship between the existence of m-units and the existence of a Minkowski unit is investigated for those n for which the class number hФ(ζn) of the n-th cyclotomic field is equal to 1. For n which is a product of two distinct primes p and q, we derive a sufficient condition for the existence of a Minkowski unit in the case when the field K contains a p-unit for every prime p, namely that every ideal contained in a finite list (see Lemma 11) is principal. This reduces the question of whether the existence of a p-unit and a q-unit implies the existence of a Minkowski unit to a verification of whether the above ideals are principal. As a corollary of this, we establish that every totally real cyclic field K of degree n = 2q, where q = 2, 3 or 5, contains a Minkowski unit if and only if it contains a 2-unit and a q-unit.

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References

  1. [1]
    T. Becker andV. Weispfenning in cooperation withH. Kredel,Gröbner Bases: A Computational Approach to Commutative Algebra, Graduate Texts in Mathematics 141, Springer-Verlag New York 1993.MATHGoogle Scholar
  2. [2]
    A. Brumer, On the group of units of an absolutely cyclic number field of prime degree.J. Math. Soc. Japan21 (1969), 357–358.MATHMathSciNetCrossRefGoogle Scholar
  3. [3]
    B. N. Delone and D. K. Faddeev,The theory of irrationalities of the third degree, Translations of Math. Monographs, Vol.10 (1964), American Mathematical Society.Google Scholar
  4. [4]
    S. Jakubec, On Divisibility of Class Number of Real Abelian Fields of Prime Conductor.Abh. Math. Sem. Univ. Hamburg63 (1993), 67–86.MATHMathSciNetCrossRefGoogle Scholar
  5. [5]
    J. Masley andH. Montgomery, Cyclotomic fields with unique factorization.J. Reine Angew. Math.286/287 (1976), 248–256.MathSciNetGoogle Scholar
  6. [6]
    H. Minkowski, Zur Theorie der Einheiten in den algebraischen Zahlkörpern.Nachr. Wiss. Ges. Göttingen (1900), 90-93 = Ges. Abh. I, 316–319.Google Scholar
  7. [7]
    W. Narkiewicz,Elementary and Analytic Theory of Algebraic Numbers, PWN -Polish Scientific Publishers Warsaw 1990.MATHGoogle Scholar
  8. [8]
    B. L. van der Waerden,Moderne Algebra, Springer 1950.Google Scholar
  9. [9]
    B. A. Zeinalov, The units of a cyclic real field (in Russian).Dagestan State Univ. Coll. Sci. Papers, Math. Phys. (1965), 21–23.Google Scholar

Copyright information

© Mathematische Seminar 1996

Authors and Affiliations

  • F. Marko
    • 1
  1. 1.Department of MathematicsCarleton UniversityCanada

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