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Annihilators for the class group of a cyclic field of prime power degree. II. (English) Zbl 1155.11054
Summary: We prove, for a field \(K\) which is cyclic of odd prime power degree over the rationals, that the annihilator of the quotient of the units of \(K\) by a suitable large subgroup (constructed from circular units) annihilates what we call the non-genus part of the class group. This leads to stronger annihilation results for the whole class group than a routine application of the Rubin-Thaine method would produce, since the part of the class group determined by genus theory has an obvious large annihilator which is not detected by that method; this is our reason for concentrating on the non-genus part.
The present work builds on and strengthens previous work of the authors [Acta Arith. 112, No. 2, 177-198 Zbl 1065.11089)]; the proofs are more conceptual now, and we are also able to construct an example which demonstrates that our results cannot be easily sharpened further.

MSC:
11R33 Integral representations related to algebraic numbers; Galois module structure of rings of integers
11R20 Other abelian and metabelian extensions
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