# Definition:P-adic Number/Quotient of Cauchy Sequences in P-adic Norm

## 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.

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 $\struct {\Q_p, \norm {\,\cdot\,}_p}$.

### Representative of a $p$-adic Number

Each Cauchy sequence $\sequence {y_n}$ of the left coset $\sequence {x_n} + \NN$ is called a **representative** of the $p$-adic number $\sequence {x_n} + \NN$.

## Field Operations on $p$-adic Numbers

By Field Operations of P-adic Numbers as Quotient of Cauchy Sequences, the field operations on $\Q_p$ are defined by:

- $+ :\quad \forall \sequence{x_n} + \NN, \sequence{y_n} + \NN \in \CC \, \big / \NN$:
- $\quad \paren{\sequence{x_n} + \NN} + \paren{\sequence{y_n} + \NN} = \sequence{x_n + y_n} + \NN$

- $\circ :\quad \forall \sequence{x_n} + \NN, \sequence{y_n} + \NN \in \CC \, \big / \NN$:
- $\quad \paren{\sequence{x_n} + \NN} \paren{\sequence{y_n} + \NN} = \sequence{x_n y_n} + \NN$

## Rational Numbers as Dense Subfield of $p$-adic Numbers

By Rational Numbers are Dense Subfield of P-adic Numbers, the mapping $\phi: \Q \to \Q_p$ defined by:

- $\map \phi r = \sequence {r, r, r, \dotsc} + \NN$

where $\sequence {r, r, r, \dotsc} + \NN$ is the left coset in $\CC \, \big / \NN$ that contains the constant sequence $\sequence {r, r, r, \dotsc}$, defines an isometric isomorphism between $\Q$ and $\map \phi \Q$.

In addition, $\map \phi \Q$ is a dense subfield of $\Q_p$.

It is natural to *identify* $\Q$ with $\map \phi \Q$.

## Also see

- Corollary to Cauchy Sequences form Ring with Unity for a proof that $\CC$ is a commutative ring of Cauchy sequences.

- Corollary to Null Sequences form Maximal Left and Right Ideal for a proof that $\NN$ is a maximal ideal of $\CC$.

- Corollary to Quotient Ring of Cauchy Sequences is Division Ring for a proof that the quotient ring $\Q_p = \CC \big / \NN$ is a field.

- Quotient Ring of Cauchy Sequences is Normed Division Ring for a proof that $\norm {\, \cdot \,}_p$ on $\Q_p$ is a norm.

- Non-Archimedean Division Ring Iff Non-Archimedean Completion for a proof that $\norm {\, \cdot \,}_p$ on $\Q_p$ is a non-Archimedean norm.

- Completion of Normed Division Ring for a proof that $\struct {\Q_p, \norm {\,\cdot\,}_p}$ is the completion of $\struct {\Q, \norm {\,\cdot\,}_p}$.

- Normed Division Ring is Field iff Completion is Field for a proof that $\struct {\Q_p, \norm {\,\cdot\,}_p}$ is a valued field.

- $p$-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.

- Embedding Division Ring into Quotient Ring of Cauchy Sequences for proof of the existence of a distance-preserving monomorphism of $\Q$ into $\Q_p$.

- Normed Division Ring is Dense Subring of Completion for a proof that the rational numbers $\Q$ can be identified with a subfield of the
**p-adic numbers**.

## Sources

- 1997: Fernando Q. Gouvea:
*p-adic Numbers: An Introduction*... (previous) ... (next): $\S 3.2$: Completions, Definition $3.2.9$

- 2007: Svetlana Katok:
*p-adic Analysis Compared with Real*... (previous) ... (next): $\S 1.4$ The field of $p$-adic numbers $\Q_p$