Uncertain about Uniformizing Elements of Elliptic Curves.

As this is homework, I'll try not to say too much. Recall the definitions:

The local ring of $C$ at $P=(\alpha,\beta)\in C(k)$ is $k[C]_{\mathfrak{p}}$, where $k[C]=k[x,y]/(y^2-x)$ and $\mathfrak{p}=(x-\alpha,y-\beta)k[C]$. Its maximal ideal is $\mathfrak{m}_P=\mathfrak{p}k[C]_{\mathfrak{p}}$. A uniformizing element of $P$ is a generator of $\mathfrak{m}_P$.

Beyond this, the exercise requires no knowledge of DVR's, only some basic facts on local rings.

Hint 1:

Show that $y-\beta$ is a uniformizing element of $P$.

Hint 2:

If $y+\beta$ is a unit in the local ring, then $x-\alpha$ is a uniformizing element of $P$.

Hint 3:

Show that $y+\beta$ is a unit in the local ring if and only if $2\beta\neq0$.

This covers the 'if' part of the exercise. The 'only if' part should not be hard once you understand the 'if' part.