$6.32 \star$ Let $\alpha$ and $\cos \alpha \pi$ both be rational numbers, find all possible values of $\cos \alpha \pi$.


Let $\alpha=\frac{p}{q}, p, q \in \mathbf{Z},(p, q)=1$, and $q \geqslant 1$. Let $\theta=\alpha \pi$, then $\cos q \theta=$ $\cos p \pi= \pm 1$. By induction, it is easy to see that $2 \cos q \theta$ is a $q$-th degree monic polynomial in $2 \cos \theta$ with integer coefficients. Let this polynomial be $(2 \cos \theta)^{q} + a_{q-1}(2 \cos \theta)^{q-1} + \cdots + a_{0}$, where $a_{i} \in \mathbf{Z}$. Then $2 \cos \theta$ is a rational root of the polynomial $x^{q} + a_{q-1} x^{q-1} + \cdots + a_{0} \mp 2$, hence $2 \cos \theta \in \mathbf{Z}$. Combining this with $|2 \cos \theta| \leqslant 2$, we find that $2 \cos \theta \in \{ \pm 2, \pm 1, 0\}$, so $\cos \theta \in \left\{ \pm 1, \pm \frac{1}{2}, 0\right\}$.

On the other hand, when $\alpha=2,1, \frac{1}{3}, \frac{2}{3}, \frac{1}{2}$, we have $\cos \theta=1,-1, \frac{1}{2},-\frac{1}{2}, 0$ respectively. Therefore, the possible values of $\cos \alpha \pi$ are $0, +1, +\frac{1}{2}$.