How to find polynomial of minimal degeree?

Question

Let $F := Q( 3^{1/3} , ω)$ where $ω $ is a primitive root of unity. (

. (a) Find a polynomial of minimal degree that has ω as a zero

I know the minimal polynomial for $ ω$ is $x^2+x+1$ but how to find it ?

$x= ω=1/2 +\sqrt ( 3)i$ Then , $x^2-x+1$=0

Thanks .

Answer 1

$\omega$ is a root of $x^2+x+1$ because $\omega$ is a root of $x^3-1=(x-1)(x^2+x+1)$ and $\omega\ne1$.

If $\omega$ were a root of a polynomial of smaller degree, then the polynomial would have degree $1$and $\omega$ would be rational, which is not.

Therefore, $x^2+x+1$ is the minimal polynomial of $\omega$.