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

Theorem
Let $\Z_p$ be the $p$-adic integers for some prime $p \ne 2$.

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

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

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

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

Proof
From Group of Units is Group:
 * $\Z_p^\times$ is a subgroup of $\Z_p$.

From Power of Elements is Subgroup:
 * $\paren{\Z_p^\times}^2$ is a subgroup of $\Z_p^\times$.