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Annihilators for the class group of a cyclic field of prime power degree. (English) Zbl 1065.11089
Let $$K/{\mathbb Q}$$ be a cyclic extension of prime power degree $$p^k$$, $$p$$ odd, which is totally and tamely ramified at $$s\geq 2$$ prime numbers $$p_1,\dots,p_s$$, and unramified outside $$S:=\{p_1,\dots,p_s\}$$. Denote by $$A_K$$ the $$p$$-part of the class group of $$K$$ and by $$E_K:={\mathbb Z}_p \otimes {\mathcal O}_K^*$$ the $$p$$-adic completion of the units of $$K$$. At least $$2$$ primes are assumed to be ramified in $$K$$, since otherwise $$A_K$$ is trivial. Let $$G =\langle\sigma \rangle$$ be the Galois group of $$K/{\mathbb Q}$$. Using annihilation theorems of F. Thaine [Ann. Math. (2) 128, No. 1, 1–18 (1988; Zbl 0665.12003)] and K. Rubin [Invent. Math. 89, 511–526 (1987; Zbl 0628.12007)], the authors show the existence of an $$S$$-unit $$\varepsilon \in K$$ (which is a unit once a certain conditions on ”minors” holds true for $$K$$) such that $$\varepsilon^{\sigma -1}$$ is a unit with the inclusion $$\text{Ann}(E_K/\langle\varepsilon^{\sigma -1} \rangle) \subset \text{Ann}((\sigma -1)A_K)$$, where the ideal annihilators Ann are relative to the ring $${\mathbb Z}_p[G]/ \langle\sum_{\tau \in G} \tau \rangle$$.
When the non-genus part $$(\sigma -1)A_K$$ of $$A_K$$ is cyclic, or when $$k=1$$, the preceding inclusion turns out to be an equality and one can also replace the two annihilators by Fitting ideals. However, in general, the inverse inclusion remains open.

##### MSC:
 11R29 Class numbers, class groups, discriminants 11R27 Units and factorization 11R20 Other abelian and metabelian extensions
##### Keywords:
class group; cyclotomic units; cyclic fields; annihilators
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