Ideals of P-adic Integers
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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:
- $I = p^k \Z_p$
$\blacksquare$