Some \(\widetilde A_ n\)-extensions obtained from generalized Laguerre polynomials
Let \(L(x,\lambda)\) be the \(n\)-th degree generalized Laguerre polynomial. It is proved that for \(n\geq 14\) the Galois group of \(f(x)=L(x,-2-n)\) is \(A_n\) if \(n\equiv 1\pmod 4\), and \(S_n\) otherwise. Moreover it is proved that the splitting field of \(f(x)\) can be embedded in a field with absolute Galois group isomorphic to \(\widetilde A_n\), the double cover of \(A_n\), if and only if \(n\equiv 1\pmod 8\).
scientific article; zbMATH DE number 729072