# Definition:Valued Field of P-adic Numbers

## Definition

Let $p$ be any prime number.

Let $\Q_p$ be the field of $p$-adic numbers.

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

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

### Field of P-adic Numbers

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

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 $\CC \, \big / \NN$.

The **field of $p$-adic numbers** is the field $\Q_p$.

For any Cauchy sequence $\sequence{x_n}$ in $\struct{\Q, \norm {\,\cdot\,}_p}$, let $\eqclass{x_n}{}$ denote the left coset of $\sequence{x_n}$ in $\Q_p$.

### $p$-adic Number

Each left coset $\eqclass{x_n}{}$ in $\Q_p$ is called a **$p$-adic number**.

### P-adic Norm on P-adic Numbers

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

- $\ds \forall \eqclass{x_n}{} \in \Q_p: \norm {\eqclass{x_n}{} }_p = \lim_{n \mathop \to \infty} \norm{x_n}_p$

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

## Also see

- P-adic Numbers form Non-Archimedean Valued Field for as proof that $\struct {\Q_p, \norm {\,\cdot\,}_p}$ is a valued field with a non-Archimedean norm.

- P-adic Numbers form Completion of Rational Numbers with P-adic Norm for a proof that that $\struct {\Q_p, \norm {\,\cdot\,}_p}$ is the completion of $\struct {\Q, \norm {\,\cdot\,}_p}$ and is unique up to isometric isomorphism.

- Rational Numbers are Dense Subfield of P-adic Numbers for a proof that $\struct {\Q, \norm {\,\cdot\,}_p}$ is isometrically isomorphic to a dense subfield of $\struct {\Q_p, \norm {\,\cdot\,}_p}$ and so $\Q$ can be identified with a dense subfield of the
**valued field of p-adic numbers**.

- Results about
**$p$-adic numbers**can be found**here**.

## 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 $\Q_p$