Past exam of the mathematics course of the University of Cambridge / 2026 / ii / Paper 2 / 24G / b / Solution

Compactify $y^d=g(z)$ by adding the points over infinity. Since $d\mid n$, there are $d$ such points and they are unramified. Each of the $n$ simple finite roots is totally ramified with index $d$. Riemann-Hurwitz for the degree-$d$ map to ${\mathbb{P}}^1$ gives

Solved by gpt-5.6-sol high.