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**Ideal class groups of Iwasawa-theoretical abelian extensions over the rational field.**
*(English)*
Zbl 1011.11072

Author’s summary: Let \(P\) be the set of all prime numbers. For each \(l \in P\) let \(\mathbb Z_t\) denote the ring of \(l\)-adic integers and, given any algebraic number filed \(F\), let \(C_F(l)\) denote the \(l\)-primary component of the ideal class group of \(F\), and \(C^{-}_F(l) \) the \(l\)-primary component of the kernel of the norm map from the ideal class group of \(F\) to that of the maximal real subfield of \(F\). Let \(S\) be a non-empty finite subset of \(P\). Denote by \(\mathbb Q\) the rational field, by \(\mathbb C\) the complex field, and by \(\mathbb Q^S\) the abelian extension over \(\mathbb Q\) in \(\mathbb C\) such that the Galois group of \(\mathbb Q^S/ \mathbb Q\) is topologically isomorphic to the additive group of the direct product \(\prod_{l \in S} \mathbb Z_t\). Let \(K\) be an imaginary finite extension over \(\mathbb Q^S\) in \(\mathbb C\) such that \(K/\mathbb Q\) is an abelian extension. In this paper, the author first proves that each of certain arithmetic progressions contains at most finitely many \(l \in P\) for which \(C^{-}_K (l)\) are nontrivial, and that the natural density of \(P\) of the set of all \(l \in P\) with \(C^{-}_K(l)=1\) is equal to \(1\). These results have been shown by L. C. Washington for the basic case where \(|S |=1 \) [Math. Ann. 214, 177-193 (1975; Zbl 0302.12007)]. The author next devotes much part of the paper to proving that, if \(S\) consists only of an odd prime \(p\), then \(C_{\mathbb Q^S}(l)= 1\) holds for every prime number \(l \geq \frac{3}{2} p^2 \log p \) which is a primitive root modulo \(p^2\).

In the last part of the paper, some additional results are given along with some related problems: among others, the last proposition states that, for any positive integer \(n\), the class number of the cyclotomic field of \(3^n\)th roots of unity is relatively prime to every \(l\in P\) with \(l\equiv 2 \) or \(5 \pmod 9\); while, to the author, it seems very likely that the natural density in \(P\) of the set of all \(l\in P \) with \(C_K(l) =1\) is equal to \(1\), and it seems a particularly interesting problem to know (in the case of \(|S |\)) whether the ideal class group of \(\mathbb Q^S\) is trivial, infinite, or of finite order greater than \(1\).

In the last part of the paper, some additional results are given along with some related problems: among others, the last proposition states that, for any positive integer \(n\), the class number of the cyclotomic field of \(3^n\)th roots of unity is relatively prime to every \(l\in P\) with \(l\equiv 2 \) or \(5 \pmod 9\); while, to the author, it seems very likely that the natural density in \(P\) of the set of all \(l\in P \) with \(C_K(l) =1\) is equal to \(1\), and it seems a particularly interesting problem to know (in the case of \(|S |\)) whether the ideal class group of \(\mathbb Q^S\) is trivial, infinite, or of finite order greater than \(1\).

Reviewer: Olaf Ninnemann (Uffing am Staffelsee)