Method for finding minimal polynomial of primitive root of 1 over $\mathbb Q$

Question

• Let $\zeta_1$ be a primitive 21st root of 1. Find the minimal polynomial of $\zeta_1$ over $\mathbb Q$.

• Let $\zeta_2 $ be a primitive 15th root of 1. Find the minimal polynomial of $\zeta_2$ over $\mathbb Q$.

I focusing on the method of finding the solution with the examples above but I'm struggling to find a clear step-by-step factorisation process.

I have the formula $X^n-1=\Pi_{d|n}\phi_d(X)$ where $\phi_n$ is the nth cyclotomic polynomial.

For $\zeta_1$, would I begin by finding the numbers relatively prime to 15, then by Euler's I know that the degree of the minimal polynomial will be 8?

Where do I go from there?

I'm pretty stuck, any help would be greatly appreciated!

Answer 1

We build the cyclotomic polynomials from the divisors:

• $\Phi_1(x) = x-1$
• $\Phi_3(x) = \dfrac{x^3-1}{\Phi_1(x)} = \dfrac{x^3-1}{x-1} = x^2+x+1$
• $\Phi_5(x) = \dfrac{x^5-1}{\Phi_1(x)} = \dfrac{x^5-1}{x-1} = x^4+x^3+x^2+x+1$
• $\Phi_7(x) = \dfrac{x^7-1}{\Phi_1(x)} = \dfrac{x^7-1}{x-1} = x^6+x^5+x^4+x^3+x^2+x+1$
• $\Phi_{15}(x) = \dfrac{x^{15}-1}{\Phi_1(x) \Phi_3(x) \Phi_5(x)} = x^8-x^7+x^5-x^4+x^3-x+1$
• $\Phi_{21}(x) = \dfrac{x^{21}-1}{\Phi_1(x) \Phi_3(x) \Phi_7(x)} = x^{12}-x^{11}+x^9-x^8+x^6-x^4+x^3-x+1$

Each cyclotomic polynomial is built using the property: $$\prod_{d \mid n} \Phi_d(x) = x^n-1$$

Rearranging: $$\Phi_n(x) = \dfrac{x^n-1}{\displaystyle\prod_{d~\text{proper divisor of}~n} \Phi_d(x)}$$

The minimal polynomial of $\zeta_1$ is $\Phi_{21}(x)$ and that of $\zeta_2$ is $\Phi_{15}(x)$.