# Determine Minimal Polynomial of Primitive 10th Root of Unity

I would like to determine the minimal polynomial of the primitive tenth root of unity (denoted $\zeta$) over $\mathbb{Q}$. I know that the polynomial is given by:

$\prod_{\text{gcd}(a,10)=1)\text{ for }a=1,...,9}(x-\zeta^a)=(x-\zeta)(x-\zeta^3)(x-\zeta^7)(x-\zeta^9)=(x-\zeta)(x-\overline{\zeta})(x-\zeta^3)(x-\overline{\zeta^3})=(x^2-2(\zeta+\overline{\zeta})+1)(x^2-2(\zeta^3-\overline{\zeta^3})+1)$

I am unsure of how to simplify from here, and would also appreciate comments concerning simpler methods to finding the polynomial in question. In particular, I know how to simplify the polynomial in terms of coefficients and degrees, but I am not sure how to simplify $\zeta+\overline{\zeta}$ and $\zeta^3+\overline{\zeta^3}$.

Thank you.