Find irreducible factor of the polynomial

Question

Find all irreducible factors of the polynomial $x^{15}-1$ over $\Bbb{Q}$.

Clearly, this is a cyclotomic polynomial.

I can find all the irreducible factors but that is very calculative:

\begin{align} \Phi_1(x)&=x-1\\ \Phi_3(x)&=\frac{x^3-1}{\Phi_1(x)}\\ \Phi_5(x)&=\frac{x^5-1}{\Phi_1(x)}\\ \Phi_{15}(x)&=\frac{x^{15}-1}{\Phi_1(x)\Phi_3(x)\Phi_5(x)}. \end{align} The last equation is very calculative. That's why I am looking for any less time consuming method.

Thanks!

Answer 1

You have $$\Phi_1(x) \Phi_5(x)= x^5-1$$

Therefore

$$\Phi_{15}(x)= \frac{x^{15}-1}{(x^5-1) \Phi_3(x)}= \frac{(x^5)^3-1}{(x^5-1)\Phi_3(x)}=\frac{x^{10}+x^5+1}{\Phi_3(x) }$$

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