# On the existence of*p*-units and minkowski units in totally real cyclic fields

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DOI: 10.1007/BF02940796

- Cite this article as:
- Marko, F. Abh.Math.Semin.Univ.Hambg. (1996) 66: 89. doi:10.1007/BF02940796

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## Abstract

Let*K* be a totally real cyclic number field of degree n > 1. A unit in*K* 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 to*m*. It is proved that*K* 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(ζ_{n}) 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.