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.
 A: 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.
A: Let $ K = \mathbf Q(i) $. The question is asking "what is the image of the homomorphism $ N :  K^{\times} \to \mathbf Q^{\times} $ given by the norm map?"
By the properties of the ring $ \mathcal O_K $, we know that primes that are $ 1 $ mod $ 4 $ are split, primes that are $ 3 $ mod 4 are inert and $ 2 $ is ramified. Thus, we may factorize any element of the principal ideal domain $ \mathcal O_K $ into prime elements, and note that the inert primes contribute even multiplicities of factors to the norm. Therefore, the image of the norm map $ N : \mathcal O_K - \{ 0 \} \to  \mathbf Z^{\times} $ is the subgroup $ H $ such that the primes that are $ 3 $ modulo $ 4 $ occur with even multiplicity. Since any element of $ K $ can be multiplied by an integer and lifted into $ \mathcal O_K $, we see that the image of the norm map $ N : K^{\times} \to \mathbf Q^{\times} $ consists of elements of $ H $ multiplied by rational squares. This is just the subgroup $ G $ of $ \mathbf Q^{\times} $ consisting of rationals in which primes that are $ 3 $ mod $ 4 $ occur with even multiplicity.
Unique factorization in $ \mathbf Z $, along with the fact that there are infinitely many primes $ 3 $ modulo $ 4 $, gives the decomposition
$$ \mathbf Q^{\times} \cong \mathbf Z/2\mathbf Z \oplus \bigoplus_{p} \mathbf Z $$
where the direct sum runs over all primes $ p $. Since there are infinitely many primes that are $ 3 $ modulo $ 4 $, the image of $ G $ under this isomorphism has infinitely many $ 2 \mathbf Z $ summands, thus the quotient group $ \mathbf Q^{\times} / G $ is isomorphic to the direct sum of countably many copies of $ \mathbf Z/2\mathbf Z $, and is not finite.
