The Galois group of xn−xn−1−cdots, three dots, centered−x−1

In this paper we prove that if n is an even integer or a prime number, then the Galois group of xn−xn−1−cdots, three dots, centered−x−1 is the symmetric group Sn. This polynomial family arises quite naturally from a kind of generalized Fibonacci sequence. In order to prove our result for n=p prime, we had to prove that xp−xp−1−cdots, three dots, centered−x−1 is irreducible in image, which seems to be a result of independent interest.