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Determine the automorphisms of $\Bbb{Q}(\zeta)$, where $\zeta=e^{2\pi i/n}$ for some some natural number $n$.

As far as I can tell, the problem reduces, since the rationals are fixed, to how it moves each root of unity, and that the automorphisms move roots to unity to other roots of unity. But I can't determine which of these are and aren't automorphisms.


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An automorphism of $\mathbb Q(\zeta)$ would send $\zeta$ to another root of the minimal polynomial of $\zeta$. Do you know what the other roots of this polynomial are and how they are related? – Dane Jan 19 '13 at 5:16
Okay, so it does send it to all of the possible values roots? So their are n automorphisms? – Kamina Jan 19 '13 at 5:21
up vote 1 down vote accepted

Well, if you know a potential automorphism $\phi$ sends $\zeta$ to $\zeta^k$ (i.e., one of the other roots of unity), you know that $$ \phi(\mathbb{Q}(\zeta)) \supset \mathbb{Q}(\zeta^k).$$ Now, can you show that $$ \phi(\mathbb{Q}(\zeta)) = \mathbb{Q}(\zeta^k)?$$ When is $$ \mathbb{Q}(\zeta) = \mathbb{Q}(\zeta^k)?$$

The cases of interest will be, is $\gcd(k,n)>1$ or is $\gcd(k,n)=1$?

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Thank you, didn't think it could be reduced to a cyclic group kinda problem. – Kamina Jan 19 '13 at 6:00
You're welcome! There's another nice approach here:… I don't think it counts as a "duplicate" when the other problem has been closed :) – Alex Jan 19 '13 at 6:05

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