Quotient Group of Quadratic Residues Modulo p of P-adic Units

Theorem
Let $\Q_p$ be the $p$-adic numbers for some prime $p \ne 2$.

Let $\Q_p^\times$ denote the set of $p$-adic units.

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

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

Then the multiplicative quotient group $\Q_p^\times / \paren{\Q_p^\times}^2$ has order $4$ with:
 * $\set{1, p, c, cp}$ as a transversal

Corollary

 * $\Q_p^\times / \paren{\Q_p^\times}^2$ is isomorphic to:
 * $\Z / 2\Z \oplus \Z / 2\Z$