Form of rational solutions to $a^2+b^2=1$? Is there a way to determine the form of all rational solutions to the equation $a^2+b^2=1$?

If you know some field theory, it's possible to find the form without much messy algebra. The condition that $a^2+b^2=1$ for rational $a$ and $b$ is equivalent to the fact that $N_{\mathbb{Q}(i)/\mathbb{Q}}(a+bi)=1$.
Since $\text{Gal}(\mathbb{Q}(i)/\mathbb{Q})\cong\mathbb{Z}/2\mathbb{Z}$, the classical statement of Hilbert's Theorem 90 implies that $a+bi=y/\tau(y)$ for some $y\in\mathbb{Q}(i)$, where $\tau$ is just the complex conjugation map in this case. So for some $m+ni\in\mathbb{Q}(i)$,
$$
a+bi=\frac{m+ni}{\tau(m+ni)}=\frac{m+ni}{m-ni}=\frac{(m^2-n^2)+(2mn)i}{m^2+n^2},
$$
which implies $a=\dfrac{m^2-n^2}{m^2+n^2}$ and $b=\dfrac{2mn}{m^2+n^2}$.
