Find the minimum, irreducible polynomial

Question

I have to find the minimum, irreducible polynomial of $$e^{\pi i/3}$$ over $\mathbb{Q}$.

I have done the following:

$e^{\pi i/3}$ is a root of the equation $x^6-1=0$, right??

$$x^6-1=(x-1)(x^5+x^4+x^3+x^2+x+1)$$

How could I continue??

Answer 1

Note that it is a root of $(x^3+1) = (x+1)(x^2-x+1)$

So its minimal polynomial is $x^2-x+1$.