Hensel's Lemma/P-adic Integers/Lemma 6

Theorem

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

Let $x \in \Z_p$.

Let $k \in \N_{>0}$.

Then:

$x \equiv 0 \pmod {p^k\Z_p} \implies \exists y \in \Z_p : x = y p^k$

Proof

Let:

$x \equiv 0 \pmod{p^k\Z_p}$

By definition of congruence modulo an ideal:

$x \in p^k\Z_p$

By definition of principal ideal:

$\exists y \in \Z_p : x = y p^k$

$\blacksquare$