Let $ w_1,...,w_{ \phi(n)}$ be the primitive $n$th roots of unity of $ t^n -1 \in \mathbb Q[t]$. Show that for each $ 1 \le i \le \phi (n)$, there exists an $ \sigma\in Aut \mathbb Q(w_1)$ satisfies $ \sigma ( w_1) = w_i$.
I know the converse is true, that is given any $ \sigma\in Aut \mathbb Q(w_1)$, $ \sigma (w_1) $ always maps to some $ w_i$. But for this direction I have trouble on construct the automorphism. Any help is appreciated.
Your $w_i$ are the roots of $\Phi_n$, the $n$-th cyclotomic polynomial. So your statement is then equivalent to the irreducibility of $\Phi_n$ over $\mathbb Q$ (if $w$ is any root of $\Phi_n$, ${\mathbb Q}[w]$ is isomorphic to the quotient ring $\frac{{\mathbb Q}[X]}{\Phi_n(X)}$).
Do you already know that $\Phi_n$ is irreducible ? If not, you might take a look here for example.