Past exam of the mathematics course of the University of Cambridge / 2025 / ii / Paper 4 / 1G (Number Theory)

For an odd prime $p$, the Legendre symbol isFor distinct odd primes $p$ and $q$, quadratic reciprocity states that

The discriminant isThe discriminant criterion for a quadratic congruence modulo an odd prime therefore says that, for $p\ne 3$, a solution exists exactly whenThe quadratic character of minus three giveswhich equals $1$ precisely for $p\equiv 1(\mathrm{mod}3)$. When $p=3$, the polynomial isso it also has a solution. Hence

The classsatisfies $X^3=-1$ for every prime. Thus a solution exists for every odd prime $p$, regardless of its residue class modulo $3$.

The cube roots of minus one modulo an odd prime are controlled byFor $p\ne 3$, the quadratic factor has roots exactly when its discriminant $-3$ is a square, namely when $p\equiv 1(\mathrm{mod}3)$. In that case there are three distinct roots; when $p\equiv 2(\mathrm{mod}3)$, only $X=-1$ remains. For $p=3$,so $X=-1$ is again the unique root. Therefore