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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.

Keywords

Primary Ideal Prime Divisor Class Number Finite Index Principal Ideal 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Mathematische Seminar 1996

Authors and Affiliations

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

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