Equivalence of Definitions of P-adic Integer
This article needs proofreading. Please check it for mathematical errors. If you believe there are none, please remove {{Proofread}} from the code.To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{Proofread}} from the code. |
Theorem
Let $\struct {\Q_p, \norm {\,\cdot\,}_p}$ be the $p$-adic numbers for some prime $p$.
The following definitions of the concept of P-adic Integer are equivalent:
Definition 1
An element $x \in \Q_p$ is called a $p$-adic integer if and only if $\norm x_p \le 1$.
The set of all $p$-adic integers is usually denoted $\Z_p$.
Thus:
- $\Z_p = \set {x \in \Q_p: \norm x_p \le 1}$
Definition 2
An element $x \in \Q_p$ is called a $p$-adic integer if and only if the canonical expansion of $x$ contains only positive powers of $p$.
The set of all $p$-adic integers is usually denoted $\Z_p$.
Thus:
- $\ds \Z_p = \set {\sum_{n \mathop = 0}^\infty d_n p^n : \forall n \in \N: 0 \le d_n < p} = \set{\ldots d_n \ldots d_3 d_2 d_1 d_0 : \forall n \in \N: 0 \le d_n < p}$
Proof
Definition 1 implies Definition 2
Let $x \in \Q_p$ such that $\norm x_p \le 1$.
From P-adic Integer is Limit of Unique P-adic Expansion, there exists a $p$-adic expansion of the form:
- $\ds \sum_{n \mathop = 0}^\infty d_n p^n$
By definition of the canonical expansion:
- $\ds \sum_{n \mathop = 0}^\infty d_n p^n$ is the canonical expansion of $x$
It follows that the canonical expansion of $x$ contains only positive powers of $p$.
$\Box$
Definition 2 implies Definition 1
Let the canonical expansion of $x$ contain only positive powers of $p$.
That is:
- $x = \ds \sum_{n \mathop = 0}^\infty d_n p^n : \forall n \in \N : 0 \le d_n < p$
Case 1 : $\forall n \in \N : d_n = 0$
Let:
- $\forall n \in \N : d_n = 0$
Then:
\(\ds x\) | \(=\) | \(\ds \sum_{n \mathop = 0}^\infty 0 * p^n\) | Definition of Canonical P-adic Expansion | |||||||||||
\(\ds \) | \(=\) | \(\ds \sum_{n \mathop = 0}^\infty 0\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 0\) |
Hence:
\(\ds \norm x_p\) | \(=\) | \(\ds \norm 0_p\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 0\) | ||||||||||||
\(\ds \) | \(<\) | \(\ds 1\) |
$\Box$
Case 2 : $\exists n \in \N : d_n > 0$
Let:
- $\exists n \in \N : d_n > 0$
Let:
- $l = \min \set {i: i \ge 0 \land d_i \ne 0}$
Hence:
- $l \ge 0$
Thus:
\(\ds \norm x_p\) | \(=\) | \(\ds p^{-l}\) | P-adic Norm of P-adic Expansion is determined by First Nonzero Coefficient | |||||||||||
\(\ds \) | \(\le\) | \(\ds p^0\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 1\) |
$\blacksquare$
Sources
- 2007: Svetlana Katok: p-adic Analysis Compared with Real ... (previous) ... (next): $\S 1.4$ The field of $p$-adic numbers $\Q_p$