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; 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 Classifications:

11R33, 11R20, 11Y40 show english descriptions Integral representations related to algebraic numbers; Galois module structure of rings of integers [See also 20C10]
Other abelian and metabelian extensions
Algebraic number theory computations 11R33 - Integral representations related to algebraic numbers; Galois module structure of rings of integers [See also 20C10]
11R20 - Other abelian and metabelian extensions
11Y40 - Algebraic number theory computations

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