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## Main Question or Discussion Point

I am trying to show that if z, z

^{2}, z^{3}, ..., z^{n}=1 are n distinct roots of x^{n}-1 in some extension field of Q (the rationals), then Gal_{Q}Q(z) (the galois group of Q(z) over Q) is abelian. Would I be wrong to say that since the galois group we're talking about here only involves an extension field with one of the roots, namely z, then the only map we could have in the group would be the identity map and therefore it is abelian? Something feels wrong about this but I'm not sure how else there would be other automorphisms in the group.