Let $\alpha(x,y) = (p(x)/q(x), y\cdot s(x)/t(x))$ be an endomorphism of the elliptic curve E given by $y^2 = x^3 +Ax+B$, where $p, q, s, t$ are polymonials such that $p$ and $q$ have no common root and $s$ and $t$ have no common root
a) Using the fact that $(x, y)$ and $\alpha(x, y)$ lie on E, show that $$\frac{(x^3 + Ax +B)s(x)^2}{t(x)^2} = \frac{u(x)}{q(x)^3}$$ for some polynomial $u(x)$ such that $q$ and $u$ have no common root.
b) Suppose $t(x_0) = 0$. Use the facts that $x^3 +Ax +B$ has no multiple roots and all roots of $t^2$ are multiple roots to show that $q(x_0) = 0$. This show that if $q(x_0) \neq 0$ then $\alpha(x_0, y_0)$ is defined.
