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On the existence of \(p\)-units and Minkowski units in totally real cyclic fields. (English) Zbl 0869.11087

Let \(K\) be a totally real cyclic number field of degree \(n>1\) with unit group \(U_K\). Put \(U^*_K=U_K/\{1,-1\}\). For any unit \(\varepsilon\) of \(K\), let \(\Gamma(\varepsilon)\) denote the subgroup of \(U_K\) generated by \(-1\) and the conjugates of \(\varepsilon\), and put \(\Gamma^*(\varepsilon)=\Gamma(\varepsilon)/\{1,-1\}\). The number \(i(\varepsilon)=[U^*_K:\Gamma^*(\varepsilon)]\) is called the index of \(\varepsilon\). A unit \(\varepsilon\) is called an \(m\)-unit if \(i(\varepsilon)\) is finite and prime to \(m\), and it is called a Minkowski unit if \(i(\varepsilon)=1\).
The author first proves that \(K\) contains an \(m\)-unit for every \(m\) prime to \(n\). He then proves that, for certain values of \(n\) not containing more than two different prime factors and under certain restrictive conditions, the field \(K\) contains a Minkowski unit if and only if it contains a \(p\)-unit for each prime \(p\) dividing \(n.\)
Reviewer: V.Ennola (Turku)

MSC:

11R27 Units and factorization
11R20 Other abelian and metabelian extensions
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