## On the existence of Minkowski units in totally real cyclic fields.(English)Zbl 1089.11062

The following result is proved: Let $$K$$ be a totally real, cyclic number field of degree $$n=6, 10$$ or $$14$$. Then $$K$$ has a Minkowski unit if and only if all the norm maps from the unit group of $$K$$ to the unit group of any subfield of $$K$$ are surjective. For this purpose, the author considers a totally real, cyclic number field $$K$$ of composite degree $$n$$ and investigates “structured” $$O$$-modules for $O = \mathbb Z [x] \Bigm/ \Bigl( \frac {x^n-1}{x-1} \Bigr),$ which reflect in some way the Galois module structure of the torsion free part of the unit group, $$U_K = E_K /\{\pm 1\}$$, of $$K$$: If $$K$$ has a Minkowski unit then $$U_K$$ is a structured $$O$$-module and isomorphic to $$O$$. Propositions 4.1, 4.2 and 5.2 give criteria when structured $$O$$-modules are isomorphic for the case that $$n$$ is the product of two primes. Specializing these ideas for $$n=6, 10$$ and $$14$$ proves the main result.

### MSC:

 11R27 Units and factorization
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### References:

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