Question: 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.

Answer: 
There's another way. You need a fourth degree divisor of x10 - 1 that doesn't have the six non-primitive roots of 1 as roots. Those are the fifth roots and -1. Thus, you divide x10 - 1 by x5 - 1 to get x5 + 1, and then divide that by x + 1 to get x4 - x3 + x2 - x + 1 .
Also, if you continue on your original path, you want to multiple everything out to a sum of monomials, and the coefficient of each monomial will simplify. For example, the coefficient of x3 will be the negative of the sum of the primitive tenth roots, which is the positive of the sum of the primitive fifth roots, which is -1. Other coefficients simplify similarly.