1. The problem statement, all variables and given/known data I'm trying to find the galois group of x^5 - 1 over Q, and then for each subgroup of the galois group identify which subfield is fixed. 2. Relevant equations 3. The attempt at a solution If w = exp(2*I*PI/5), then the roots not in Q are w, w^2, w^3, w^4. Its fairly easy to see by checking possible automorphisms that the Galois group is isomorphic to the multiplicative group of integers mod 5, which is in turn isomorphic to C4. There may be a mistake there already.. but I'm not sure. Here comes the confusion- C4 has one non-trivial proper subgroup, in this case found by multiplying only the elements w and w^4 together. But there are no intermediate fields as far as I can tell, since each root is primitive and so generates the other roots. Please help! I've been teaching myself this course so I think I've probably just misunderstood something somewhere but not sure what..