P-adic Norm not Complete on Rational Numbers/Proof 2/Lemma 4

Theorem
Let $p, x_1, k \in \Z_{\gt 0}: p \nmid x_1, p \nmid k$

Let $a \in \Z$ be any integer.

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

Let $f' \paren{X} \in \Z [X]$ be the formal derivative of $f \paren{X}$.

Then:
 * $\map {f’} {x_1} \not \equiv 0 \pmod p$

Proof
By Euclid's Lemma for Prime Divisors then:
 * $p \nmid kx_1^{k-1}$

Hence:
 * $kx_1^{k-1} \not \equiv 0 \mod p$

The formal derivative $f' \paren{X} \in \Z [X]$ of $f \paren{X}$ is by definition:
 * $kX^{k-1}$

Then:
 * $\map {f’} {x_1} = kx_1^{k-1} \not \equiv 0 \pmod p$