$\mathsf{Spec} \: \mathbb{Q}[X,Y]/(XY^2-m)\cong \mathsf{Spec} \: \mathbb{Q}[X,1/X]$

Question

How I can prove that $\mathsf{Spec} \: \mathbb{Q}[X,Y]/(XY^2-m)$ is isomorphic (as scheme) to $\mathsf{Spec} \: \mathbb{Q}[X,1/X]$?

This is an example from Algebraic Geometry and Aritmethic Curves - Qing Liu.

Answer 1

In general, we have $R[T]/(fT-1) \cong R[\frac{1}{f}]$. In our case we get

$$\mathbb Q[X,Y]/(XY^2-m) = \mathbb Q[X,Y]/(\frac{Y^2}{m}X-1) \cong \mathbb Q[Y,\frac{m}{Y^2}] = \mathbb Q[Y,\frac{1}{Y^2}] = \mathbb Q[Y,\frac{1}{Y}].$$