Hensel's Lemma/P-adic Integers/Lemma 6
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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:
- $\exists y \in \Z_p : x = y p^k$
$\blacksquare$