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:
 * $\quad \set{0}$
 * $\quad \forall k \in \N$, $p^k\Z_p$

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:

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:

Case 2 : $b = 0$
We have:

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

It follows that:
 * $I \subseteq p^k \Z_p$

By the definition of set equality:
 * $I = p^k \Z_p$