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 invertible elements of $\Q_p$.

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

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

Proof
By definition of field:
 * $\Q_p^\times = \Q_p \setminus \set{0}$ is an abelian group

From Group of Units is Group:
 * $\struct{\Q_p^\times, \times}$ is a subgroup of $\struct{\Q_p^*, \times}$

From Power of Elements is Subgroup:
 * $\struct{\paren{\Q_p^\times}^2, \times}$ is a subgroup of $\struct{\Q_p^\times, \times}$

By definition of quotient group, the quotient group $\Q_p^\times \mathop/ \paren{\Q_p^\times}^2$ can be formed.

$\dfrac p 1 = p \not\equiv a^2 \pmod p (\forall a \in \Q_p^\times)$

$\dfrac c p \not\equiv a^2 \pmod p$

$\dfrac c 1 = c \not\equiv a^2 \pmod p$