Characterization of P-adic Unit has Square Root in P-adic Units/Condition 2 implies Condition 3

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 $u$.

Let $c_0$ be a quadratic residue of $p$.

Then:

$\exists y \in \Z_p : y^2 \equiv u \pmod{p\Z_p}$

Proof

From Partial Sum Congruent to P-adic Integer Modulo Power of p:

$u \equiv c_0 \pmod {p\Z_p}$

Let $c_0$ be a quadratic residue of $p$.

By definition of quadratic residue:

$\exists x_0 \in \Z : x_0^2 \equiv c_0 \pmod {p\Z_p}$

Then:

$x_0^2 \equiv u \pmod {p\Z_p}$

$\blacksquare$