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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.

11R29 Class numbers, class groups, discriminants
11R27 Units and factorization
11R20 Other abelian and metabelian extensions
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