Proving that $\mathbb{Q}(\omega)$ is the splitting field for $f(x)=x^4 - x^2 + 1$ over $\mathbb{Q}$

Question

I have been given that $\omega = e^{\pi i\over{6}}$, a 12th root of unity. I have shown that $\omega$ is a root of the polynomial $f(t)=t^4 -t^2 +1$, as are $\omega^5,\omega^7,\omega^{11}$ and that $f$ is the minimal polynomial of $\omega$ over $\mathbb{Q}$. Now I must show that $$\Gamma(\mathbb{Q(\omega):Q})$$ has order 4 (and subsequently that it is isomorphic to the Klein 4-group). My immediate thought is to show that $\mathbb{Q(\omega)}$ is the splitting field of $f$ over $\mathbb{Q}$ and hence $\mathbb{Q(\omega):Q}$ is a normal extension then use the fundamental theorem of Galois Theory, but I am unsure how to prove this. Any advice is welcome.

Answer 1

As $\mathbb{Q}(\omega)$ is a field, i.e. closed under multiplication, then $\omega^5,\omega^7,\omega^{11}\in\mathbb{Q}(\omega)$ and hence the splitting field of $f$: $$\mathbb{Q}(\omega,\omega^5,\omega^7,\omega^{11})=\mathbb{Q}(\omega)$$