Complex factorization via roots of unity

Roots. The roots of $X^n-1$ are $\omega_k=e^{2ik\pi/n}$ for $k=0,1,\ldots,n-1$.

Factorization.
$$X^n-1=\prod_{k=0}^{n-1}(X-\omega_k).$$

Over $\mathbb{R}$. Pair conjugates $\omega_k$ and $\omega_{n-k}$ (for $k\neq0,n/2$) to get quadratics $X^2-2\cos(2k\pi/n)X+1$. The factors are:
- $X-1$;
- if $n$ even, $X+1$;
- for $k=1,\ldots,\lfloor(n-1)/2\rfloor$, quadratics $X^2-2\cos(2k\pi/n)X+1$.

Conclusion. Over $\mathbb{C}$, $X^n-1$ splits into $n$ linear factors; over $\mathbb{R}$, into linear factors for the real roots ($\pm1$) and irreducible quadratics.