Equivalence of Definitions of P-adic Integer

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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

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