Let $E : y^2 = x^3 + ax + b$ with $a,b\in \Bbb Z$ be an elliptic curve.
If $(x, y) ∈ E(\Bbb Q)$ Prove $\exists \alpha, \beta\, \gamma \in \Bbb Z$ such that $x = \alpha /\gamma ^2 , y = \beta /\gamma ^3$ with $(\alpha \beta , \gamma) = 1$.
I don't see how to tackle this problem.
Thank you for your help.
Let $p$ be a prime dividing the denominator of $x$. Suppose $p^r$ is the largest power of $p$ dividing it. Then $p^{3r}$ is the largest power of $p$ dividing the denominator of $x^3+ax+b=y^2$. This means that $p$ also divides the denominator of $y$, and if $p^s$ is the largest power of $p$ dividing its denominator then $p^{2s}$ is the largest power of $p$ dividing the denominator of $y^2$. Therefore $2s=3r$ which means that $(r,s)=(2t,3t)$ for some positive integer $p$.
Likewise if $p$ does not divide the denominator of $x$ it cannot divide the denominator of $y$.
So the denominators of $x$ and $y$ can be written as $p_1^{2t_1}p_2^{2t_2}\cdots p_k^{2t_k}$ and $p_1^{3t_1}p_2^{3t_2}\cdots p_k^{3t_k}$ respectively, therefore as $\gamma^2$ and $\gamma^3$ where $\gamma=p_1^{t_1}p_2^{t_2}\cdots p_k^{t_k}$.