= Ramification at infinity of a superelliptic covering
{title2=$w^r=z^n-1,\quad d=\gcd(n,r),\quad e_\infty=r/d$}
In the compactified <Riemann surface> of $w^r=z^n-1$, a positive circuit enclosing the $n$ simple finite zeros sends the sheet label $j$ to $j+n$ modulo $r$. This permutation has $d=\gcd(n,r)$ cycles of length $r/d$. Each cycle fills in one point over infinity, with ramification index $r/d$. Hence infinity is unramified exactly when $r$ divides $n$. Each finite zero has ramification index $r$, so the base sphere has exactly $n$ branch values in that case and $n+1$ otherwise.
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