Characterization of P-adic Unit has Square Root in P-adic Units

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

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

Let $u \in Z_p^\times$ be a $p$-adic unit.

Let $u = c_0 + c_1p + c_2p^2 + \ldots$ be the $p$-adic expansion of $u$.


 * $(1)\quad \exists x \in \Z_p^\times : x^2 = u$
 * $(2)\quad c_0$ is a quadratic residue of $p$
 * $(3)\quad \exists y \in \Z_p : y^2 \equiv u \pmod{p\Z_p}$
 * $(3)\quad \exists y \in \Z_p : y^2 \equiv u \pmod{p\Z_p}$

Proof
From Partial Sum Congruent to P-adic Integer Modulo Power of p:
 * $u \equiv c_0 \pmod {p\Z_p}$