On the ideal class groups of the maximal cyclotomic extensions of algebraic number fields
Let \(k_0\subset \mathbb{C}\) be a number field. Let \(k_{\infty}\) be the maximal cyclotomic extension of \(k_0\) and \(C_{\infty}\) its class group; \(C_{\infty}\) is a direct sum of its \(p\)-primary components \(C_{\infty}(p)\) for all prime number \(p\): \(C_{\infty}=\bigoplus_pC_{\infty}(p)\). Let \(k_1/k_0\) be the subextension of \(k_{\infty}/k_0\) obtained by adjoining a primitive \(4\)-th root of unity and all primitive \(l\)-th root of unity, where \(l\) is an odd prime number. Put \(\mathfrak{g}=\operatorname{Gal}(k_{\infty}/k_1)\), then \(\mathfrak{g}\) acts naturally on \(C_{\infty}\). The purpose of the paper is to investigate the structure of \(C_{\infty}\) as a \(\mathfrak{g}\)-module. From now on suppose that \(k_0\) is totally real and \(p\) is an odd prime number. Let \(k_{\infty}^{+}\) denote the maximal totally real subfield of \(k_{\infty}\). The complex conjugation \(\rho\), which is a generator of \(\operatorname{Gal}(k_{\infty}/k_{\infty}^{+})\), acts on \(C_{\infty}(p)\). Let \(C_{\infty}(p)^{\pm}=\{c\in C_{\infty}(p), \rho(c)=\pm c\}\). Then \(C_{\infty}(p)=C_{\infty}(p)^{+} \oplus C_{\infty}(p)^{-}\), as discrete \(\mathfrak{g}\)-modules. According to a result of \textit{M. Kurihara} [J. Eur. Math. Soc. (JEMS) 1, No. 1, 35--49 (1999; Zbl 0949.11055)], \(C_{\infty}(p)^{+}=\{0\}\). Therefore \(C_{\infty}(p)=C_{\infty}(p)^{-}\). Let \(W(p)\) be the group of all \(p\)-power roots of unity. For a pro-\(p\) \(\mathfrak{g}\)-module \(X\), \(\Hom(X, W(p))\) denotes the set of continuous homomorphisms from \(X\) to \(W(p)\). The group \(\mathfrak{g}\) acts on \(\Hom(X, W(p))\) by: \(\sigma(f)(x)=\sigma(f(\sigma^{-1}(x))\), so that \(\Hom(X, W(p))\) is a discrete \(\mathfrak{g}\)-module. Let \(\mathcal{A}_p\) be the completed group algebra of \(\mathfrak{g}\) over \(\mathbb{Z}_p\) (the ring of \(p\)-adic integers), which is a pro-\(p\) \(\mathfrak{g}\)-module. Let \(\mathfrak{C}_p=\Hom(\mathcal{A}_p, W(p))\). The main result of the paper is: as discrete \(\mathfrak{g}\) modules, \(C_{\infty}(p)\simeq \bigoplus_{N=1}^{\infty}\mathfrak{C}_p\). The proof uses a characterization of the pro-\(p\) \(\mathfrak{g}\) module \(\prod_{N=1}^{\infty}\mathcal{A}_p\) in terms of the solvability of embedding problems.
scientific article; zbMATH DE number 6383636