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 $Z_p^+$ be the set of $p$-adic units.

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

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

Then:
 * $\exists x \in \Z_p : x^2 = u$


 * $c_0$ is a quadratic residue of $p$.