\(2\)-birationality propagation (Q2847834)

Let \(l\) be a prime integer and \(K\) be a number field; we say that \(K\) is \(l\)-rational number field if the Galois group \(\mathrm{Gal}(M/K)\) of the maximal pro-extension \(M/K\) , which is \(l\)-ramified and \(\infty\)-split, is free pro-\(l\)-group. We say that K is \(l\)-birational number field if it is rational for 2 places over \(l\) and \(K\) contains the \(l\)-roots of unity. Let \(L/K\) be a 2-birational CM-extension of totally real 2-rational number field. In this paper, the authors characterize in terms of tame ramification totally real 2-extensions \(K^{'}/K\) such the compositum \(L^{'} = LK^{'}\) is still 2-birational. In the case where \(K^{'}/K\) is linearly disjoint from the cyclotomic \(\mathbb Z_{2}\)-extension \(K^{c}/K\), the authors prove that \(K^{'}/K\) is at most quadratic. Furthermore, they construct infinite towers of such 2-extensions.