I know from reading that the Galois group of a cyclotomic polynomial is isomorphic to $(\mathbb{Z}/n\mathbb{Z})^*$. While I believe this, I can't figure out why that should work. In particular, it's not obvious to me that all of the available permutations of the roots should actually extend to automorphisms, because the way that we think of the action $\overline{m}\cdot\zeta^k=\zeta^{mk}$ is not bijective over the whole field and in any case it does not fix $\mathbb{Q}$.
Are there explicit expressions of the automorphisms that look pretty, or do we generally argue their existence in some other way?
(Tagged homework because I am trying to get an insight into a problem on an outstanding assignment in my class)
Every automorphism is completely determined by where $\zeta$ is sent to. $\zeta$ can be sent to any other primitive $n$th root of unity by the very basics of field theory (any other root of the $n$th cyclotomic polynomial). So any automorphism is determined by some $k$ where $\zeta \rightarrow \zeta^k$, where $(k,n)=1$ and the isomorphism between the galois group and $(\Bbb Z/n\Bbb Z)^*$ should be clear.