Algebraic Curves: Valuation at a point

Question

I would like to understand the notion of valuation on the local ring of a curve at a point.

In the Book The Arithmetic of Elliptic Curves in chapter 2, Example 1.3 $$V:\ Y^{2}=X^{3}+X$$

I don't understand how they get the following answer:

$\mbox{Ord}_{(0,0)}(Y)=1$ and $\mbox{Ord}_{(0,0)}(X)=2$

Thanks

Answer 1

In the local ring of $V$ at $(0,0)$ we have that $x=\frac{y^2}{1+x^2}$. This shows that the maximal ideal in the local ring at $(0,0)$ is generated by $y$ (really the image of $y$ in the local ring).

This means that $\mbox{ord}_{(0,0)}y=1$ and since $x$ is a unit multiplied by $y^2$ in the local ring, $\mbox{ord}_{(0,0)}x=2$.