# Definition:P-adic Number/P-adic Norm Completion of Rational Numbers

## 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.

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**.

## Notes

By Completion of Valued Field, for each prime $p \in \Z$ there exists a field $\Q_p$ with a non-Archimedean norm $\norm {\,\cdot\,}_p$ such that:

- $i) \quad$there exists a distance-preserving monomorphism $\phi:\struct{\Q,\norm {\,\cdot\,}_p} \to \struct{\Q_p,\norm {\,\cdot\,}_p}$.

- $iii) \quad \struct{\Q_p,\norm {\,\cdot\,}_p}$ is complete.

The valued field $\struct{\Q_p,\norm {\,\cdot\,}_p}$ is unique (up to isometric isomorphism).

Furthermore, by Normed Division Ring is Dense Subring of Completion the mapping $\phi: \Q \to \map \phi \Q$ is an isometric isomorphism onto the dense subfield $\map \phi \Q$ of $\Q_p$.

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

## Also see

- $p$-adic Norm is Non-Archimedean Norm for a proof that $\struct {\Q, \norm {\,\cdot\,}_p}$ 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.

- Normed Division Ring is Dense Subring of Completion for a proof that $\struct {\Q, \norm {\,\cdot\,}_p}$ is isometrically isomorphic to a dense subfield of $\struct {\Q_p, \norm {\,\cdot\,}_p}$.

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

## Sources

- 1997: Fernando Q. Gouvea:
*p-adic Numbers: An Introduction*... (previous) ... (next): $\S 3.3$: Exploring $\Q_p$