On locally cyclotomic towers (Q1305356)

Let \(K\) be an algebraic number field. The logarithmic \(\ell\)-class group \(\widetilde{\mathcal C\ell}_K\) of \(K\) can be identified via \(\ell\)-adic class field theory with the Galois group \(\text{Gal}(K^{lc}/K^c)\), where \(K^{lc}\) is the maximal abelian locally cyclotomic pro-\(\ell\)-extension of \(K\) and \(K^c\) is the cyclotomic \(\mathbb Z_\ell\)-extension \(\mathbb Q^c\cdot K\). The purpose of the article is to investigate the existence of a finite extension \(N/K\) such that \(\widetilde{\mathcal C\ell}_N=0\). The authors construct canonically a tower of fields \((K_n)_{n\in\mathbb N}\) such that each \(K_n\) is a locally cyclotomic Galois \(\ell\)-extension of \(K\). Put \(K_\infty=\cup_{n\in\mathbb N}K_n\). They show that, if \(K_\infty/K\) is finite, then \(K_\infty\) is in a sense the smallest such field \(N\), whereas in the opposite case no such \(N\) exists.