Assume that $p$ is a prime number congruent to 1 modulo 4, and fix $j\in\mathbb{Z}/p^\alpha\mathbb{Z}$ such that $j^2=-1$. Let $a\in\mathbb{Z}/p^\alpha\mathbb{Z}$ such that $p$ does not divide $a$. Show that $(a,b)\in S(p^\alpha)$ if and only if
$$b = ja \quad \text{or} \quad b = -ja.$$