Proving Abelian Property of Galois Group in Q(z) over Q

[jeffreydk]

I am trying to show that if z, z², z³, ..., zⁿ=1 are n distinct roots of xⁿ-1 in some extension field of Q (the rationals), then Gal_QQ(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.

 

[Hurkyl]

To say that Gal_Q Q(z) is abelian requires an imlicit assumption: that Gal_Q Q(z) actually exists. For that to happen. Q(z) must actually be a Galois extension. Also, if Q(z) is a Galois extension of Q, then #Gal_Q Q(z) = [Q(z) : Q] -- so if Gal_Q Q(z) consists only of the identity, then z must actually be a rational number... which is only true if n < 3.

The problem secretly told you that Q(z) is a Galois extension of Q, and you know that automorphisms of Q(z) map roots of z^n - 1 to roots of z^n - 1. You made an unwarranted assumption that led you to conclude that the only automorphism of Q(z) is trivial -- what was it?


If you're still confused, maybe it's worth first working out the Galois group of the entire splitting field of x^n - 1, rather than just of Q(z).

 

[jeffreydk]

Yes that makes sense that its a Galois extension, because we're told that each of the roots are distinct (and there are n of them) so it is a splitting field and has characteristic 0, thus it's Galois. Although your argument makes perfect sense, saying that |Gal_QQ(z)|=[Q(z):Q]=1 if the group is trivial, I am having trouble noticing where my unwarranted assumption was, because to me, if the field is only extending z and not the rest of the roots, there are no combinations of isomorphisms that can be made other than some f:z-->z. I'll try working out the group for the entire splitting field.

 

[Hurkyl]

Yes that makes sense that its a Galois extension, because we're told that each of the roots are distinct (and there are n of them) so it is a splitting field and has characteristic 0, thus it's Galois.

The splitting field is Galois -- but is Q(z)? Typically, adjoining a single root of a polynomial does not give a Galois extension, because you need to include all of its conjugates to make the extension Galois.

if the field is only extending z and not the rest of the roots,

Actually, that is the unwarranted assumption.