Ideals of P-adic Integers

Theorem

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

Then the ideals of $\Z_p$ are the principal ideals:

$\text a) \quad \set 0$

$\text b) \quad \forall k \in \N: p^k \Z_p$

Corollary

$\Z_p$ is a principal ideal domain

Proof

Let $\nu_p: \Q_p \to \Z \cup \set {+\infty}$ denote the $p$-adic valuation on the $p$-adic numbers.

Let $\Z_p^\times$ denote the $p$-adic units.

Let $I \ne \set 0$ be a non-null ideal of $\Z_p$.

Hence:

$\exists j \in I : \map {\nu_p} j < \infty$

Let:

$k = \inf \set {\map {\nu_p} i : i \in I}$

Hence:

$k \le j < \infty$

Let:

$a \in I : a \ne 0 \land \map {\nu_p} a = k$

From P-adic Number is Power of p Times P-adic Unit:

$\exists u \in \Z_p^\times : a = p^k u$

We have:

\(\ds p^k\)

\(=\)

\(\ds u^{-1} a\)

\(\ds \)

\(\in\)

\(\ds I\)

Definition of Ideal of Ring

\(\ds \leadsto \ \ \)

\(\ds \paren {p^k}\)

\(=\)

\(\ds p^k\Z_p\)

Definition of Principal Ideal

\(\ds \)

\(\subseteq\)

\(\ds I\)

Definition of Ideal of Ring

Let $b \in I$.

Case 1 : $b \ne 0$

Let:

$w = \map {\nu_p} b$

Then:

$k \le w < \infty$

From P-adic Number is Power of p Times P-adic Unit:

$\exists u' \in \Z_p^\times : b = p^w u'$

We have:

\(\ds b\)

\(=\)

\(\ds p^w u'\)

\(\ds \)

\(=\)

\(\ds p^k \cdot p^{w - k} u'\)

As $k \le w$

\(\ds \)

\(\in\)

\(\ds p^k \Z_p\)

Definition of Principal Ideal and $p^{w - k} u' \in \Z_p$

$\Box$

Case 2 : $b = 0$

We have:

\(\ds b\)

\(=\)

\(\ds 0\)

\(\ds \)

\(=\)

\(\ds p^k \cdot 0\)

\(\ds \)

\(\in\)

\(\ds p^k \Z_p\)

Definition of Principal Ideal and $0 \in \Z_p$

$\Box$

In either case: $b \in p^k \Z_p$

It follows that:

$I \subseteq p^k \Z_p$

By the definition of set equality:

$I = p^k \Z_p$

$\blacksquare$