zbMATH — the first resource for mathematics

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.

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