Let $p$ be a prime number $\geq 2$ and $\zeta := \zeta_{p}$ a complex number where $\zeta \neq 1$ and $\zeta^p=1$, i.e., is a $p$th root of unity. This is a question from a brazillian book, Introdução à Algebra (Introduction to Algebra), by Gonçalves. A. This question is posted before the chapter about groups and galois theory, so it is purely an splitting field exercise:
Show that $\mathbb{Q}[\zeta_{p}] = \{a_{0}, a_{1}\zeta, \dots, a_{p-2}\zeta^{p-2}, a_{i} \in \mathbb{Q}\}$.
This comes right after a pretty similar question, where the polynomial that we look around is $x^5 -1$. I could not do this example either, even algebrically, writing down the solutions (using complex form) and trying to understand why the $\zeta^4$ root is gone. The maximum that I got is that $\zeta^4 = \bar{\zeta}$, but I could not find a way to span it whit his "smaller brothers". So, any help with this is going to be great!
FYI, I know about galois theory, but this is not intended to be looked with this "deeper" eyes.
Thanks in advance!