TITLE: The group $\mathbb{Q}^\times / \left( \mathbb{Q}^\times \right)^2$
QUESTION [6 upvotes]: Does there exist a nice description of the group $\mathbb{Q}^\times / \left(  \mathbb{Q}^\times \right)^2$? What is its order?

REPLY [7 votes]: Here, we see that $\mathbb Q^{\times}\cong \mathbb Z/2\mathbb Z\oplus\left(\bigoplus_{i=1}^{\infty}\mathbb Z\right)$. Continuing to write additively, $$2\left(\mathbb Z/2\mathbb Z\oplus\left(\bigoplus_{i=1}^{\infty}\mathbb Z\right)\right)=\bigoplus_{i=1}^\infty\mathbb 2\mathbb Z.$$ More precisely, the multiplication by $2$ turns a tuple $(a,b,c,\dots)$ with almost all terms zero into the tuple $(0,2b,2c,\dots)$.
It follows that $$\mathbb Q^\times/2\mathbb Q^\times\cong\mathbb Z/2\mathbb Z\oplus\bigoplus_{i=1}^\infty\mathbb Z/2\mathbb Z\cong\bigoplus_{i=1}^\infty\mathbb Z/2\mathbb Z.$$