Central extension of \(S_n\) as Galois groups of regular extensions of \(\mathbb{Q}(T)\) (Q1175753)

The following theorem is proved: Every finite central extension of the symmetric group \(S_n\), \(n\in \mathbb{N}\), is realizable as the Galois group of a regular extension of \(\mathbb{Q}(t)\). The proof being based on \textit{J. F. Mestre}'s construction of \(\tilde {A}_n\)-extensions [J. Algebra 131, 483--495 (1990; Zbl 0714.11074)]\ finalizes a number of partial results by the author, N. Vila and others.