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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
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