This was a problem from an old qualifying exam, that I solved today, with a few pointers. First of all, is it reducible? It actually is. Note that . , as a prime element of that divides not must divide the polynomial, the Galois group of which we are looking for. The other factor of it corresponds to the multiplicative group of , which has elements. Seeing that it has elements of order and elements of order and is abelian, it must be . Thus, the answer is .