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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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