I'm trying to understand a proof that galois extension of $x^n-1$ over $\mathbb{Q}$ is isomorphic to $(\mathbb{Z}_n)^\times$.
I can see why there is an injection of $\text{Gal}(x^n-1)$ in $(\mathbb{Z}_n)^\times$. However, when it comes to proving the surjectivity, I am confused.
Given $\zeta$ a primitive $n$th root of unity and $r \in (\mathbb{Z}_n)^\times$. Why is it sufficient to show that $\zeta$ and $\zeta^r$ have the same minimal polynomial ? Isn't it possible that for $\zeta$ and $\zeta^r$ to have the same minimal polynomial without having an automorphisme $\tau$ such that $\tau(\zeta) = \zeta^r$ ?
No. If $\alpha$ and $\beta$ have the same minimal polynomial $p(X)$ over $K$, then $K[\alpha]\cong K[\beta] \cong K[X]/(p(X))$.
It is a basic result of Galois theory that this isomorphism extends to an automorphism of the algebraic closure, though that is not even necessary in this case.