# Odd-degree hyperellptic curve ramified at infinity

Let $C$ be the projective curve with the affine model given by the equation $Y^2=F(X)$, where $F$ is a polynomial in $x$ with degree $d$ over a field $k$ with characteristic not equal to 2. When $d$ is odd, why is $C$ ramified at the point at infinity?

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## 1 Answer

Hurwitz's formula says
$$2g(C)-2=2\cdot (2g(\mathbb P^1)-2)+ \mid Ram\mid$$
Hence the number $\mid Ram\mid$ of ramification points is even and since there are only $d$ in the affine part, there must be one at infinity too.

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