# Automorphisms and the roots of unity.

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.

Thanks

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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? – Chanler Jan 19 '13 at 5:21

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 $\gcd(k,n)=1$?