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 $\Q_p$