Characterization of Integer has Square Root in P-adic Integers

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

Let $a \in Z$ be an integer such that $p \nmid a$.

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


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

That is, an integer $a$ not divisible by $p$ has a square root in $\Z_p$ ($p \ne 2$) $a$ is a quadratic residue of $p$.

Proof
Let $F \in \Z[X]$ be the polynomial:
 * $\map F X = X^2 - a$

By definition of formal derivative the formal derivative of $F$ is:
 * $\map {F'} X = 2X$