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

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(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.