Uncertain about Uniformizing Elements of Elliptic Curves.

Uncertain about Uniformizing Elements of Elliptic Curves.

pI am following a subject on Elliptic Curves and have come accross the
notion of a uniformizer. Wikipedia tells me that an element is a
uniformizer of a Discrete Valuation Ring, if it generates the (only)
maximal ideal. This seems sort of clear, but I have no idea how to apply
it to elliptic curves. Consider the following question:/p pLet $k$ a
field, $C: y^2=x$ a smooth curve in $\mathbb{A}^2$ and $P=(\alpha,\beta)$
a point in $C(k)$. Furthermore suppose that the characteristic of $k\neq
2$. Show that $x-\alpha$ is a uniformizing element of $P$ if and only if
$P\neq (0,0)$./p pNow this is not even intuitively clear to me. The ideal
we want to look at is $(y-\beta,x-\alpha)$ I suppose, since this maps
$k[x,y]/(y^2-x)$ to $0\in k$, but how do I show that
$(y-\beta,x-\alpha)=(x-\alpha)$ iff $P\neq (0,0)$?/p pI also cannot find
any information about such problems anywhere (I have the book Rational
points on elliptic curves by Silvermann, but it has nothing about
uniformizers)./p pI would appreciate some explanation (or a solution with
an explanation so I can apply this to other problems) or a reference to a
book which explains this to somebody who has not heard about Discrete
Valuation Rings or Uniformizers before./p pEDIT: This is still not clear
to me, I tried finding info in the recommended book, but it still doesn't
offer enough information. Could anybody be so helpful to explain how to
find uniformizers for such functions?/p