Definition:P-adic Norm/P-adic Numbers
Definition
Let $p$ be a prime number.
Let $\norm {\,\cdot\,}_p$ denote the $p$-adic norm on the rationals $\Q$.
Let $\Q_p$ be the field of $p$-adic numbers.
That is, $\Q_p$ is the quotient ring of the ring of Cauchy sequences over $\struct {\Q, \norm {\,\cdot\,}_p}$ by null sequences in $\struct {\Q, \norm {\,\cdot\,}_p}$.
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 $\Q_p$.
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 $\Q_p$.
Notation
Since the $p$-adic norm $\norm {\,\cdot\,}_p$ on $p$-adic Numbers $\Q_p$ may be considered an extension of the $p$-adic norm $\norm {\,\cdot\,}$ on the rational numbers $\Q$ there is generally no need to distinguish the two norms as the context is usually sufficient to distinguish them.
So the notation $\norm {\,\cdot\,}_p$ is used for both norms.
This is similar to the use of the absolute value $\size {\,\cdot\,}$ on the standard number classes.
Also see
- $p$-adic Norm on Rational Numbers is Non-Archimedean Norm for a proof that $\norm {\,\cdot\,}_p$ is a non-archimedean norm on $\Q$ and the pair $\struct {\Q, \norm {\,\cdot\,}_p}$ is a valued field.
- $p$-adic Norm not Complete on Rational Numbers for a proof that $\struct {\Q, \norm {\,\cdot\,}_p}$ is not a complete valued field.
- $p$-adic Numbers form Non-Archimedean Valued Field for as proof that $\norm {\,\cdot\,}_p$ is a non-Archimedean norm.
- Rational Numbers are Dense Subfield of $p$-adic Numbers for a proof that the $p$-adic norm on the $p$-adic numbers may be considered an extension of the $p$-adic norm on the rational numbers.
Sources
- 1997: Fernando Q. Gouvea: p-adic Numbers: An Introduction: $\S 3.2$ Completions, Definition $3.2.11$
- 2007: Svetlana Katok: p-adic Analysis Compared with Real: $\S 1.4$ The field of $p$-adic numbers $\Q_p$