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 $\map f X \in \Z \sqbrk X$ be the polynomial $X^k - a$

Let $\map {f'} X \in \Z \sqbrk X$ be the formal derivative of $\map f X$.

Then:

$\map {f'} {x_1} \not \equiv 0 \pmod p$

Proof

By Euclid's Lemma for Prime Divisors:

$p \nmid k x_1^{k - 1}$

Hence:

$k x_1^{k - 1} \not \equiv 0 \mod p$

The formal derivative $\map {f'} X \in \Z \sqbrk X$ of $\map f X$ is by definition:

$k X^{k - 1}$

Then:

$\map {f'} {x_1} = k x_1^{k - 1} \not \equiv 0 \pmod p$

$\blacksquare$