Is every extension of \(\mathbb{Q}\) the specialization of a branched covering?
It is shown that every Galois extension of the rationals with symmetric group \(S_ n\) can be obtained as a specialization of a suitable model of a Galois branched covering \({\mathcal C}\to P^ 1\) with the same Galois group, having at most \(2n-1\) branch points. (Here \({\mathcal C}\) is a connected complete and smooth curve over the complex field.) A similar result is obtained for finite abelian extensions of any number field. The proof is based on the work of \textit{D. Saltman} [Adv. Math. 43, 250-283 (1982; Zbl 0484.12004)].
scientific article; zbMATH DE number 562445