Hensel's Lemma/P-adic Integers/Lemma 7

Theorem

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

Let $T$ be the set of $p$-adic digits.

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

Then:

$x \in \Z_p \implies \exists y \in T : y p^k \equiv x p^k \pmod {p^{k+1}\Z_p}$

Let $x \in \Z_p$.

$x = \ds \sum_{n \mathop = 0}^\infty d_n p^n$

We have:

$\ds x - d_0$

$=$

$\ds \paren{d_0 + d_1p + d_2p^2 + d_3p^3 + \ldots } - d_0$

$\ds $

$=$

$\ds \paren{d_1p + d_2p^2 + d_3p^3 + \ldots }$

$d_0$ terms cancel

$\ds $

$=$

$\ds p \paren{d_1 + d_2p + d_3p^2 + \ldots }$

Factor $p$ from each term

$\ds $

$\in$

$\ds p\Z_p$

$d_1 + d_2p + d_3p^2 + \ldots$ is the $p$-adic expansion of a $p$-adic integer

$\ds \leadsto \ \ $

$\ds xp^k - d_0p^k$

$=$

$\ds \paren{x - d_0}p^k$

$\ds $

$\in$

$\ds p^{k+1}\Z_p$

$\ds \leadsto \ \ $

$\ds xp^k$

$\equiv$

$\ds d_0p^k \pmod {p^{k+1}\Z_p}$

Let $y = d_0$.

The result follows.

$\blacksquare$