How to find the index of this subgroup of $\mathbb{Q}^{\times}$? Consider the following subgroup $H$ of the multiplicative group $\mathbb{Q}^{\times}$:
\begin{equation}
H=\{a^{2}+b^{2}\:|\:a,b\in\mathbb{Q}\}-\{0\}.
\end{equation}
Is there any easy of computing the index $(\mathbb{Q}^{\times}\colon H)$? Or at least saying if it is finite.
I can't really post my attempt because I have no idea how to start. Any tips or references? Thanks.

We can write $q\in\Bbb Q^\times$ as $\pm2^{a_2}3^{a3}5^{a_5}7^{a_7}\cdots$ with almost all $a_p=0$.
The map $$\Bbb Q^\times\to\bigoplus_{n=1}^\infty\Bbb Z/2\Bbb Z,\quad q\mapsto(\operatorname{sgn}(q),a_3\bmod2,a_7\bmod 2, a_{11}\bmod 2,\ldots) $$
where we run over all primes $\equiv 3\pmod 4$ is a group epimorphism having $H$ in its kernel. Consequently, $[\Bbb Q^\times:H]$ is infinite.
