Let's say $\omega$ is a primitive $6^{\rm th}$ root of unity. Then we know $[\mathbb{Q}(\omega):\mathbb{Q}]=\varphi(6)=2$. If I count the number of linearly independent basis elements for $\mathbb{Q}(\omega)$ from the set of $\{1, \omega, \omega^{2}, \omega^{3}, \omega^{4}, \omega^{5}\}$, I should be able to construct $4$ equations. 3 of them are:
- $1+\omega+\omega^{2}+\omega^{3}+\omega^{4}+\omega^{5}=0$
- $1+\omega^{2}+\omega^{4}=0$
- $1+\omega^{3}=0$
What is the $4^{\rm th}$ equation?
It's really just $1-\omega+\omega^2=0$ which is a factor of all the other polynomials of $\omega$ you listed. An arbitrary element of $\mathbb Q[\omega]$ can be uniquely written as $a+b\omega$ where $a,b\in\mathbb Q$.