P-adic Norm not Complete on Rational Numbers/Proof 2/Lemma 3

Theorem
Let $p \ge 3$ be a prime number.

Let $x_1 \in \Z_{\gt 0}: \dfrac {p + 1} 2 \le x_1 \lt p$

Let $a = x_1^2 + p$

Let $f \paren{X} \in \Z [X]$ be the polynomial:
 * $X^2 - a$

Then:
 * $\map f {x_1} \equiv 0 \pmod p$