Find the degree of a field extension

Question

Suppose $F$ is the minimal subfield in $\mathbb{C}$ containing all the roots of polynomial $x^4-x^2+1$. Find the degree of a field extension $[F:\mathbb{Q}]$.
I understand that I need to find the degree of a minimal polynomial in $F$ in order to get the answer. But I don't really understand what to do. Am I supposed to find all the roots of the given polynomial over $\mathbb{C}$, and then if these roots are $x_1,\dots ,x_k$, find $[\mathbb{Q}(x_1,\dots ,x_k):\mathbb{Q}]$?

Answer 1

$$p(x):=x^4-x^2+1\implies x^2_{1,2}=\frac{1\pm\sqrt 3\,i}2=e^{\pm\pi i/3}\implies x_{1,2,3,4}=\pm w_1:=\pm e^{\pi i/6},\,\pm w_2:=\pm e^{-\pi i/6}$$

Thus, the splitting field of $\;p(x)\;$ is $\;\Bbb Q(w_1,w_2)\;$ ...Can you take it from here?

Further hint: You take out one of $\;w_1,\,w_2\;$ in the above field ...can you see why?