Past exam of the short options course of the University of Oxford / 2023 / 1 / c / ii / Solution

For $f(x)=x^2-n$, a root modulo the odd prime $p$ is nonzero because $p\nmid n$. Its derivative $2x$ is therefore nonzero modulo $p$. Part (i) gives exactly one lift of each root. A nonzero quadratic residue modulo an odd prime has two distinct roots, $x$ and $-x$; a nonresidue has none. Hence