Let $p$ be prime. What is the biggest number $m$ such that $\{1,\zeta_p,\zeta_p^2,...,\zeta_p^m\}$ is linearly independent over $\mathbb{Q}$? Let $p$ be prime. What is the biggest number $m$ such that $\{1,\zeta_p,\zeta_p^2,...,\zeta_p^m\}$ is linearly independent over $\mathbb{Q}$? Here, $\zeta_p$ is a primitive $p$th root of unity, such as $e^{\frac{2\pi i}{p}}$
Here is my attempt:
So I know I can factor $x^p-1=(x-1)(x^{p-1}+x^{p-2}+...+x+1)$. Then I would try to do a quotient
$\frac{\mathbb{Q}[x]}{(x^{p-1}+x^{p-2}+...+x+1)}\cong\mathbb{Q}(\zeta_p)$. 
Thus, everything on the left can be written as a linear combination of from the set $\{1,x,...,x^{p-2}\}$. I conclude that $m=p-2$ is the right number, since in my mind I identified $x$ with $\zeta_p$.
Is this correct?

The set $\{1,\zeta_p,\zeta_p^2,\dots,\zeta^{p-2}\}$ is a basis for $\Bbb Q(\zeta_p)$ over $\Bbb Q$.
On the other hand, $1+\zeta_p+\dots\zeta_p^{p-1}=0$, so $\{1,\zeta_p,\dots,\zeta_p^{p-1}\}$ is dependent.
