Show that if $\zeta =e^{\frac{2\pi i}{n}}$, then $\mathbb{Q}(\zeta)=\{q_0+q_1 \zeta + \dots + q_{n-1}\zeta^{n-1}: q_i \in \mathbb{Q}\}$.

Question

Show that if $\zeta =e^{\frac{2\pi i}{n}}$, then $\mathbb{Q}(\zeta)=\{q_0+q_1 \zeta + \dots + q_{n-1}\zeta^{n-1}: q_i \in \mathbb{Q}\}$.

My question: The most frustrating part is to show $\frac{1}{q_0+q_1 \zeta + \dots + q_{n-1}\zeta^{n-1}}$ also has the form $q_0+q_1 \zeta + \dots + q_{n-1}\zeta^{n-1}$, which I struggle to show. Any hint would be greatly appreciated.