Q:

Square classes of $\mathbb{Q}_2^\times$?

Does anyone know how to easily compute them? We know that a number is a square modulo $2^k$ if and only if it's a square modulo $8$. This gives a bunch of integers that represent square classes. I also know that
$$\mathbb{Q}_2^\times \cong \mathbb{Z}\times (1+2\mathbb{Z}_2),$$
but I can't figure out how to find the square classes in $1+2\mathbb{Z}_2$. Is there some really simple solution for this? I can't seem to figure out how to apply Hensel's lemma here.

A:

By your observations, a square element of $1+2\mathbb{Z}_2$ must actually live in $1+8\mathbb{Z}_2$.  So 
$$
\mathbb{Q}_2^\times/\mathbb{Q}_2^{\times 2}\approx \mathbb{Z}/2\mathbb{Z}\times \frac{1+2\mathbb{Z}_2}{1+8\mathbb{Z}_2}\approx \mathbb{Z}/2\mathbb{Z}\times \mathbb{Z}/2\mathbb{Z}\times \mathbb{Z}/2\mathbb{Z}.
$$
The first factor of $\mathbb{Z}/2\mathbb{Z}$ corresponds to choosing even/odd-ness of the power of 2 dividing the element of $\mathbb{Q}_2^\times$, the latter two any choice of coset represenatives for odd squares mod 8.  So one possible enumeration of representatives for these 8 classes are the classes of
$$
\{\pm 1,\pm 2,\pm 5,\pm 10\}.
$$

