On the Existence of p-Units and Minkowski Units in Totally Real Cyclic Fields

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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 hℚ(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 g-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.

Original languageEnglish (US)
Pages (from-to)89-111
Number of pages23
JournalAbhandlungen aus dem Mathematischen Seminar der Universitat Hamburg
Volume66
Issue number1
DOIs
StatePublished - Jan 1 1996

All Science Journal Classification (ASJC) codes

  • Mathematics(all)

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