Splittings of \(\widetilde{\mathbb Q} /\mathbb Q\)
Let \(K\) be a subfield of a field \(M.\) \(M\) is said to be PAC over \(K,\) if for every absolutely irreducible variety \(V\) of dimension \(r\geq 0\) and for each dominating separable rational map \(\varphi:V\to {\mathbb A}^r\) over \(M,\) there exists \(a\in V(M)\) such that \(\varphi (a)\in K^r.\) It is proved that if \(M\) is a PAC field over a subfield \(K\) and \(N\) is a separable algebraic extension of \(M\) then there exists a separable algebraic extension \(L\) of \(K,\) linearly disjoint from \(M,\) such that \(LM=N.\) Results are used from [\textit{M. Jarden} and \textit{A. Razon}, Isr. J. Math. 86, No.~1-3, 25--59 (1994; Zbl 0802.12007)].
scientific article; zbMATH DE number 1462954