Computing the degree of an isogeny

Let $E$ be an elliptic curve with a $p$-torsion point. Denote this point by $P$. Why does is isogeny $\phi: E \rightarrow E/\langle P \rangle$ of degree $p$?

I do know that if $\phi$ is separable, then the degree is the order of the kernel which is $\#\langle P\rangle = p$. This gives the right answer but I'm not sure on how to show separability.

-
One way to show separability is by looking at the associated extension of function fields. What is the function field of $E/\langle P \rangle$ (considered as a subfield of $K(E)$ via $\phi$) and how is it constructed? – Thom Tyrrell Dec 19 '12 at 20:36