Maximal 2-extensions with restricted ramification (Q5925816)

Let \(K\) be a number field, and let \(S\) be a finite set of finite prime divisors of \(K\). Let \(G _S\) be the Galois group of the maximal \(p\)-extension unramified outside \(S\). The case \(S\) containing the primes above \(p\) has been extensively studied by Koch, Wingberg, Neukirch and Shafarevich among many others, and the structure of \(G _S\) is fairly well understood. On the other hand, when \(S\) contains no primes above \(p\), very little is known concerning the structure of \(G _S\). NEWLINENEWLINENEWLINEWhen \(K = {\mathbb{Q}}\), \(p = 2\) and \(S\) is a finite set of odd rational primes, then allowing ramification at \(\infty\) and denoting by \({\mathbb{Q}} _S\) the maximal \(2\)-extension of \({\mathbb{Q}}\) unramified outside \(S \cup \{\infty\}\), we have that \(G _S = \text{ Gal}({\mathbb{Q}} _S/ {\mathbb{Q}})\) is finite cyclic when \(|S|= 1\), \(G _S\) is infinite if \(|S|\geq 4\) as a consequence of the Golod-Shafarevich class field tower theorem [\textit{E. S. Golod} and \textit{I. R. Shafarevich}, Izv. Akad. Nauk. SSSR, Ser. Mat. 28, 261-272 (1964; Zbl 0136.02602); Transl., II. Ser., Am. Math. Soc. 48, 91-102 (1965; Zbl 0148.28101)]. When \(|S|= 3\), \(G _S\) can be either finite of infinite. It is unknown whether \(G _S\) is finite of infinite in general when \(|S|= 2\). NEWLINENEWLINENEWLINEIn the paper under review, the authors give, for the case \(|S|= 2\), the explicit structure of \(G _S\) for three different families.