A note on free pro-\(p\)-extensions of algebraic number fields. II
For a rational prime \(p\) and an algebraic number field \(k\), a Galois extension \(K/k\) whose Galois group is isomorphic to the free pro-\(p\)-group on \(d\) generators will be called a free pro-\(p\)-extension of rank \(d\), or an \(F_d\)-extension. Let \(\rho= \rho(k)\) be the maximal rank of all free pro-\(p\)-extensions of \(k\). Leopoldt's conjecture immediately implies that \(\rho\leq 1+r_2\), where \(r_2\) is the number of complex places of \(k\). But examples exist for which \(\rho<1 +r_2\) [see Part I, J. Théor. Nombres Bordx. 5, 165-178 (1993; Zbl 0784.11052)]. Besides, an \(F_\rho\)-extension is not necessarily unique. In this short note, improving upon an observation of Iwasawa, the author shows that if \(k\) admits an \(F_{1+r_2}\)-extension \(K\) which contains the cyclotomic \(\mathbb{Z}\)-extension of \(k\), then \(K\) is unique.
scientific article; zbMATH DE number 980692