The paper has the following relation: $$y^{(p-1)/p_i} \equiv \alpha^{x(p-1)/p_i} \equiv \gamma_i^x \equiv \gamma_i^{b_0} \pmod p$$
where $\gamma_i = \alpha^{(p-1)/p_i}$. I understand this relation and can show how each step is derived. However I'm stuck at the next step. From the paper:
$\gamma_i$ is primitive $p_i$th root of unity. There are therefore only $p_i$ possible values for $y^{(p-1)/p_i} \pmod p$, and the resultant value uniquely determines $b_0$
What exactly does this mean and how do I get $b_0$ from $\gamma_i$?