Characterization of Integer has Square Root in P-adic Integers/Sufficient Condition

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

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

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

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

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:
 * $\map {F'} X = 2X$

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

By definition of quadratic residue of $p$:
 * $\exists b \in \Z : a \equiv b^2 \pmod p$

Then:
 * $\map F b = b^2 - a \equiv 0 \pmod p$

and
 * $\map {F'} b = 2b$

By hypothesis:
 * $p \nmid 2$

and
 * $p \nmid b^2$

From the contrapositive statement of Divisor Divides Multiple:
 * $p \nmid b$

From the contrapositive statement of Euclid's Lemma for Prime Divisors:
 * $p \nmid 2b$

Hence:
 * $\map {F'} b = 2b \not\equiv 0 \pmod p$

From Congruence Modulo Equivalence for Integers in P-adic Integers:
 * $\map F b \equiv 0 \pmod {p\Z}$

and
 * $\map {F'} b \not\equiv 0 \pmod {p\Z}$

From Hensel's Lemma for P-adic Integers:
 * $\exists x \in \Z_p : \map F x = 0$

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