Determine Minimal Polynomial of Primitive 10th Root of Unity

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 1

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 .

Answer 2

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.