Definition:P-adic Norm/P-adic Numbers

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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


Sources