I'm currently studying elliptic curves and have the following problem:
Let E be an elliptic curve over $\mathbb{Q}$ defined by $y^2 +y = x^3 - x$. Let P = (0,0) be a point on the curve.
i) show that $x$ is a uniformizer for $E$ at $P$.
ii) compute $ord_P(y)$
iii) compute the divisor of $y+x^2$
Now I've got the following questions:
i) In general, how do I show that something is a uniformizer? I guess showing that $u(P) = 0$ and every function $f(x,y)$ can be written in the form $ f = u^r*g$ with $r \in \mathbb{Z}$ and $g(P) \neq 0 $ would satisfy the definition. Also showing that $x$ generates the maximal ideal of the local ring $\mathcal{O}_{E,P} = \{ \frac{g}h\in K(E) \ \vert \ h(P) \neq 0 \} \cup \{0\}$ would be another way to do it, but I don't know how to approach this. (Would it be enough, just to show that the line $x$ is not tangent to the curve?)
iii) Here I have to compute the zeroes and poles of $g := y+x^2$ ant then write $div(g)$ according to our definiton $div(g) = \sum v_P(g)P $. But in this part, I really struggle to get the $v_P(g)$-part. Is there a straightforward way to compute this for each $P$?