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

Theorem

Let $x_1, p, k \in Z_{\gt 0}$ be any positive integers.

Let $a = x_1^k + p$

Let $\map f X \in \Z \sqbrk X$ be the polynomial:

$X^k - a$

Then:

$\map f {x_1} \equiv 0 \pmod p$

Proof

\(\ds \map f {x_1}\)

\(=\)

\(\ds x_1^k - \paren {x_1^k + p}\)

\(\ds \)

\(=\)

\(\ds \paren {x_1^k - x_1^k} - p\)

\(\ds \)

\(=\)

\(\ds -p\)

\(\ds \)

\(\equiv\)

\(\ds 0 \pmod p\)

$\blacksquare$