# Equivalence of Definitions of P-adic Integer

## 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 $\ds$ $\le$ $\ds p^0$ $\ds$ $=$ $\ds 1$

$\blacksquare$