Hensel's Lemma/P-adic Integers/Lemma 10
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Theorem
Let $\Z_p$ be the $p$-adic integers for some prime $p$.
Then:
- $\forall x \in \Z_p: p^k x \equiv 0 \pmod{p^{k+1}\Z_p} \implies x \equiv 0 \pmod{p\Z_p}$
Proof
We have:
\(\ds p^k x\) | \(\equiv\) | \(\ds 0 \pmod{p^{k+1}\Z_p}\) | ||||||||||||
\(\ds \leadsto \ \ \) | \(\ds p^k x\) | \(\in\) | \(\ds p^{k+1}\Z_p\) | Definition of Congruence Modulo an Ideal | ||||||||||
\(\ds \leadsto \ \ \) | \(\ds \exists y \in \Z_p: \, \) | \(\ds p^kx\) | \(=\) | \(\ds p^{k+1}y\) | Definition of Principal Ideal | |||||||||
\(\ds \leadsto \ \ \) | \(\ds \exists y \in \Z_p: \, \) | \(\ds x\) | \(=\) | \(\ds py\) | Divide by $p^k$ | |||||||||
\(\ds \leadsto \ \ \) | \(\ds x\) | \(\in\) | \(\ds p\Z_p\) | Definition of Principal Ideal | ||||||||||
\(\ds \leadsto \ \ \) | \(\ds x\) | \(\equiv\) | \(\ds 0 \pmod{p\Z_p}\) | Definition of Congruence Modulo an Ideal |
$\blacksquare$