# Definition:P-adic Number

## Contents

## Definition

Let $p$ be any prime number.

Let $\norm {\,\cdot\,}_p$ be the p-adic norm on the rationals $\Q$.

By P-adic Norm is Non-Archimedean Norm then $\norm {\,\cdot\,}_p$ is a non-archimedean norm on $\Q$ and the pair $\struct {\Q, \norm {\,\cdot\,}_p}$ is a valued field.

### $p$-adic Norm Completion of Rational Numbers

The **p-adic numbers**, denoted $\struct {\Q_p, \norm {\,\cdot\,}_p}$, is the unique (up to isometric isomorphism) non-Archimedean valued field that completes $\struct {\Q, \norm {\,\cdot\,}_p}$.

Each element of $\Q_p$ is called a **$p$-adic number**.

### Quotient of Cauchy Sequences in $p$-adic Norm

Let $\CC$ be the commutative ring of Cauchy sequences over $\struct {\Q, \norm {\,\cdot\,}_p}$.

Let $\NN$ be the set of null sequences in $\struct {\Q, \norm {\,\cdot\,}_p}$.

Let $\Q_p$ denote the quotient ring $\CC \, \big / \NN$.

Let $\norm {\, \cdot \,}_p:\Q_p \to \R_{\ge 0}$ be the norm on the quotient ring $\Q_p$ defined by:

- $\displaystyle \forall \sequence {x_n} + \NN: \norm {\sequence {x_n} + \NN }_p = \lim_{n \mathop \to \infty} \norm{x_n}_p$

By Corollary to Quotient Ring of Cauchy Sequences is Normed Division Ring, then $\struct {\Q_p, \norm {\, \cdot \,}_p}$ is a valued field.

Each left coset $\sequence {x_n} + \NN \in \CC \, \big / \NN$ is called a **$p$-adic number**.

The **p-adic numbers** is the valued field $\struct {\Q_p, \norm {\,\cdot\,}_p}$.

### $p$-adic Norm on $\Q_p$

The norm $\norm {\,\cdot\,}_p$ on $\Q_p$ is called the **$p$-adic norm** on $\Q_p$.

## Also denoted as

The **$p$-adic numbers** are often simply denoted as $\Q_p$ where the $p$-adic Norm is implied.

## Also see

- $p$-adic Norm not Complete on Rational Numbers for a proof that $\struct {\Q, \norm {\,\cdot\,}_p}$ is not a complete valued field.

- Completion Theorem for a proof that the completion of $\struct {\Q, \norm {\,\cdot\,}_p}$ exists and is unique up to isometric isomorphism.

- Completion of Normed Division Ring for a proof that $\struct {\mathcal C \big / \mathcal N, \norm {\,\cdot\,}_p}$ is the completion of $\struct {\Q, \norm {\,\cdot\,}_p}$

## Sources

- 1997: Fernando Q. Gouvea:
*p-adic Numbers: An Introduction*: $\S 3.2$: Completions - 2007: Svetlana Katok:
*p-adic Analysis Compared with Real*: $\S 1.4$ The field of $p$-adic numbers $\Q_p$