Automorphisms and the roots of unity.

Question

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

Answer 1

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