On the Galois Groups of Some Recursive Polynomials

We show that for some recursive sequence (cm)m≥1 of integers and for sufficiently large n, the Galois group of polynomial fn(x) = xn/ n! + cn−1 xn−1 /(n−1)! +· · ·+c1/x 1! + 1, contains the alternating group An. In case n is a prime number, this group is the full symmetric group Sn.