Could someone please explain me the following?
It says in the notes I am reading that $$ Spec \ (\mathbb{Q}[x,y] / (x^2 + y^2-1) ) \rightarrow Spec \ \mathbb{Q}[m] $$ given by $$ f : (x,y) \rightarrow y/(x-1) $$ for $x \not = 1$ and $$ g : m \rightarrow ((m^2-1)/(m^2 + 1), -2m/(m^2 + 1)) $$ away from $m^2 + 1 = 0$, gives that the two schemes are birational.
This is related to the classical formula of Pythagorean triples. I am confused with how to show that these maps are birational. Could someone please explain me how this works?
Thank you!
ps I am quite confused with how $f$ and $g$ are defined as well... I am not even sure if these maps are supposed to be for the schemes or for the coordinate rings?