CommeUnJeu — Math as a game

Let $m, n \in \mathbb{N}^*$ and $d = \gcd(m, n)$ (the integer gcd in $\mathbb{Z}$). The goal is to prove the identity $$ (X^n - 1) \wedge (X^m - 1) = X^d - 1 \qquad \text{in } \mathbb{C}[X] \text{ (and in } \mathbb{R}[X]\text{)}. $$

Show that for any $n \in \mathbb{N}^*$, the complex roots of $X^n - 1$ are the $n$-th roots of unity $\{e^{2 i \pi k / n} : k = 0, 1, \dots, n - 1\}$, all simple. (You may cite the chapter Polynomials.)
Show that $e^{2 i \pi k / n}$ is also an $m$-th root of unity iff $n \mid k m$.
Deduce that the common complex roots of $X^n - 1$ and $X^m - 1$ are exactly the $d$-th roots of unity in $\mathbb{C}$.
Conclude that $(X^n - 1) \wedge (X^m - 1) = X^d - 1$ in $\mathbb{C}[X]$, and that the same identity holds in $\mathbb{R}[X]$.

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