Some generalized Laguerre polynomials whose Galois groups are the alternating groups (Q584315)

The author considers one kind of generalized Laguerre polynomials \[ F_n = (-1)^n n!\sum^n_{m=0}\binom{2n}{n-m}\frac{(-X)^m}{m!}\in\mathbb Z[X] \] and shows that their Galois groups over \(\mathbb Q\) are the alternating group \(A_n\) if \(F_n\) is irreducible over \(\mathbb Q\) and \(n\) is even. He is able to prove that \(F_n\) is irreducible over \(\mathbb Q\) if \(n=2p^k\), \(p\ge 5\) a prime or \(n=4p^k\), \(p\ge 11\) a prime. These results are almost in the sense of I. Schur who gave the first families of rational polynomials whose Galois groups are \(A_n\) in cases \(n\equiv 0\pmod 4\) or \(n\equiv 1\pmod 2\).