Characterization of P-adic Unit has Square Root in P-adic Units/Condition 1 implies Condition 2

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 there exist $x \in \Z_p^\times$ such that $x^2 = u$.

Then:

$c_0$ is a quadratic residue of $p$

Proof

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

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

Let there exist $x \in \Z_p^\times$ such that $x^2 = u$

Then: $x^2 \equiv c_0 \pmod {p\Z_p}$

Let $x = x_0 + x_1p + x_2p^2 + x_3p^3 + \ldots$ be the $p$-adic expansion of $x$.

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

$x \equiv x_0 \pmod {p\Z_p}$

Then:

$x_0^2 \equiv x^2 \equiv c_0 \pmod {p\Z_p}$

By definition of quadratic residue:

$c_0$ is a quadratic residue of $p$

$\blacksquare$