Definition:P-adic Number
This page does not define the p-adic numbers as such, just provides some justification for their definition, which then happens on a different page. Hence it is confusing to someone trying to find out what they are. Basically, this page is trying to do too much.
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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:
- $\ds \forall \sequence {x_n} + \NN: \norm {\sequence {x_n} + \NN }_p = \lim_{n \mathop \to \infty} \norm{x_n}_p$
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 $\Q_p$