In this paper we present several polynomial time algorithms for Galois groups. We show: (i) There are polynomial time algorithms to determine: (a) If the Galois group of an irreducible polynomial over Q is a p-group. (b) the prime divisors of the order of a solvable Galois group (ii) Using the classification theorem for finite simple groups, there is a polynomial time algorithm to determine whether an irreducible polynomial over Q has Galois group Sn or An.