Congruence Modulo Equivalence for Integers in P-adic Integers

Theorem

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

For any $a, b \in \Z_p$ and $n \in \N$, let $x \equiv y \pmod{p^n \Z_p}$ denote congruence modulo the principal ideal $p^n\Z_p$.

For any integers $a, b \in \Z$ and $n \in \N$, let $x \equiv y \pmod{p^n}$ denote congruence modulo integer $p^n$.

Let $x, y \in \Z$ be integers.

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

The following statements are equivalent:

$(1)\quad x \equiv y \pmod{p^k \Z_p}$

$(2)\quad x \equiv y \pmod{p^k}$

$(3)\quad p^k \divides \paren{x - y}$

Proof

Lemma

$\forall a \in \Z: \dfrac a {p^k} \in \Z_p \iff \dfrac a {p^k} \in \Z$

$\Box$

We have:

\(\ds x\)

\(\equiv\)

\(\ds y \pmod{p^k\Z_p}\)

\(\ds \leadstoandfrom \ \ \)

\(\ds x - y\)

\(\in\)

\(\ds p^k\Z_p\)

Definition of Congruence Modulo an Ideal

\(\ds \leadstoandfrom \ \ \)

\(\ds \exists z \in \Z_p: \, \)

\(\ds x - y\)

\(=\)

\(\ds p^kz\)

Definition of Principal Ideal

\(\ds \leadstoandfrom \ \ \)

\(\ds \dfrac {x - y} {p^k}\)

\(\in\)

\(\ds \Z_p\)

Divide both sides by $p^k$

\(\ds \leadstoandfrom \ \ \)

\(\ds \dfrac {x - y} {p^k}\)

\(\in\)

\(\ds \Z\)

Lemma

\(\ds \leadstoandfrom \ \ \)

\(\ds \exists a \in \Z: \, \)

\(\ds \dfrac {x - y} {p^k}\)

\(=\)

\(\ds a\)

\(\ds \leadstoandfrom \ \ \)

\(\ds \exists a \in \Z: \, \)

\(\ds x - y\)

\(=\)

\(\ds p^ka\)

Multiply both sides by $p^k$

\(\ds \leadstoandfrom \ \ \)

\(\ds p^k\)

\(\divides\)

\(\ds \paren{x - y}\)

Definition of Divisor of Integer

\(\ds \leadstoandfrom \ \ \)

\(\ds x\)

\(\equiv\)

\(\ds y \pmod{p^k}\)

Definition of Congruence Modulo Integer

$\blacksquare$