## 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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### References:

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