A remark on Boston's question concerning the existence of unramified \(p\)-extensions
Let \(K\) be a number field, \(p\) an odd prime, \(K(p)\) the Hilbert \(p\)-class field of \(K\), and assume that the class number of \(K(p)\) is divisible by \(p\). \textit{N. Boston} [J. Number Theory 42, 285-291 (1992; Zbl 0768.11044)] asked whether there always exists a nontrivial unramified extension \(M/K\) such that \(\text{Gal}(M/K)\) is a \(p\)-group with the same exponent as \(\text{Gal} (K(p)/K)\), and remarked that the truth of the Fontaine-Mazur conjecture implies an affirmative answer if \(K\) has an infinite \(p\)-class field tower. Although such an extension \(M\) does not always exist [see \textit{N. Boston}, Some cases of the Fontaine-Mazur conjecture. II, J. Number Theory (to appear)], the author provides some support for the existence of \(M\) in the semisimple case by proving the following theorem, using the methods he developed in [Manuscr. Math. 79, 379-390 (1993; Zbl 0806.11053)]: Let \(p\) and \(\ell\) be distinct odd primes, and let \(k\) be a number field with finite unit group and class number prime to \(p\). Assume that \(K\) is an abelian \(\ell\)-extension such that \(p\) divides the class number of \(K\). If the order \(f\) of \(p\) modulo \(\ell\) is even, then the class number of \(K(p)\) is also divisible by \(p\), and there exists an unramified non-abelian \(p\)-extension \(M/K\) with exponent \(p\).
scientific article; zbMATH DE number 883834