Quotient Group of Quadratic Residues Modulo 2 of 2-adic Units

Theorem
Let $\Q_2$ be the $2$-adic numbers.

Let $\Q_2^\times$ denote the set of $2$-adic units.

Let $\paren{\Q_2^\times}^2 = \set{a^2 : a \in \Q_2^\times}$

Let $c \in \Q_2^\times \setminus \paren{\Q_2^\times}^2$

Then the multiplicative quotient group $\Q_2^\times / \paren{\Q_2^\times}^2$ has order $8$ with:
 * $\set{1, -1, 5, -5, 2, -2, 10, -10}$ as a transversal

and
 * $\Q_2^\times / \paren{\Q_2^\times}^2$ is isomorphic to $\Z / 2\Z \oplus \Z / 2\Z \oplus \Z / 2\Z$