I was aked to find the splitting field of polynomial like $x^4+1, x^6+1, x^4-2$, etc. The roots of unity started appearing and I had an idea.
A polynomial $x^a+c$ (lets suppose $c\in \mathbb{Q}$ and $a\in \mathbb{Z}$) is gonna have $a$ roots, right? These roots are always equally distributed in the complex plane (that is, the angle between two consecutive ones is gonna be $\pi/a$.
When $c=1$ the first root can be thought of being $e^{i\pi/a}$, because $(e^{i\pi/a})^a = e^{i\pi} = -1$ and then this root solves $x^a=-1$
When $c=-1$ the first root can be thought of being $e^{i2\pi/a}$, because $(e^{i2\pi/a})^a = e^{i2\pi} = 1$ and then this root solves $x^a=1$
In both cases, all the other roots are gonna be powers of the first root I described. In the cases where $c\neq \pm 1$, it is just a constant times $\pm 1$ and the same rules apply, but the firts root is gonna be just the first roof of unity times a constant.
So, when we wanna take the splitting field of any of these polynomails $x^a+c$ I described, we wanna find
$$\mathbb{Q}(r_1, r_2, \cdots r_a)$$
But since the $r_2,\cdots r_a$ roots are just powers of $r_1$, is it true that
$$\mathbb{Q}(r_1, r_2, \cdots r_a) = \mathbb{Q}(r_1)$$
?
Then,
$$[\mathbb{Q}(r_1, r_2, \cdots r_a):\mathbb{Q}] = [\mathbb{Q}(r_1):\mathbb{Q}]$$
?
This intuition came from this answer of another question I did: https://math.stackexchange.com/a/2439495/166180
Not quite. The problem is that for general $c$, the other roots are not powers of $r_1$. Rather, the other roots have the form $r_1\omega^n$, where $\omega=e^{2\pi i/a}$. So the splitting field can be described as $\mathbb{Q}(r_1,\omega)$, but not necessarily as $\mathbb{Q}(r_1)$.
For a very concrete example, consider $x^4-2$. You can take the first root $r_1$ to be the positive real root $\sqrt[4]{2}$. But $\mathbb{Q}(r_1)$ is then contained in $\mathbb{R}$, so cannot be the entire splitting field since not all the roots are real. Instead the splitting field is $\mathbb{Q}(r_1,i)$, where $i=e^{2\pi i/4}$ is a primitive $4$th root of unity.