$F$-Automorphism of $F(w)$ induces automorphism on $\{1,w,w^2,...,w^{n-1} \}$ only if $w$ primitive n-th root?

Question

I just stumble across a theorem in Galois theory, that says that if F is a field of characteristic p and $w$ is a primitive n-th root of unity with $p \nmid n$ than if we take a $F$-automorphism $\sigma$ of $F(w)$, then $\sigma$ induces an automorphism $\sigma_0$ of $\{1,w,w^2,...,w^{n-1}\}$ by restricting $\sigma$ to this set. But I was wondering : What could go wrong if $w$ was just an algebraic element over $F$ and not a primitive root of unity?