In an early example in Guillot's "A Gentle Course in Local Class Field Theory", he considers $\mathbb{Q}^{\times}=\{\pm1\}A$, where A is a free abelian group, basis the set of prime numbers. He then goes on to state that $\mathbb{Q}^{\times}/\mathbb{Q}^{\times2}=\{\pm1\}A/2A$.
I'm stuck seeing where this $A/2A$ is coming from, any help/hints?
The positive rational numbers can be written in the form of a finite product $\prod_ip_i^{a_i}$ with $a_i\in\Bbb Z$.
This corresponds to the finitely supported sequence $(a_i)$, that is an element of the free Abelian group $A=\bigoplus^\omega \Bbb Z$, which is embedded into $\{\pm1\} \times A$ as $(+1,(a_i))$.
A negative rational number corresponds to some $(-1,(a_i))$ in $\{\pm1\} \times A$.
Finally, the square of a nonzero rational number is always positive, and all the exponents are even numbers (as the double of the original exponent), so they correspond to $(+1,(2a_i))$.