Definition:P-adic Number/P-adic Norm Completion of Rational Numbers

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Definition

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.


The p-adic numbers, denoted $\struct {\Q_p, \norm {\,\cdot\,}_p}$, is the unique (up to isometric isomorphism) non-Archimedean valued field that completes $\struct {\Q, \norm {\,\cdot\,}_p}$.


Notes

By Completion of Valued Field, for each prime $p \in \Z$ there exists a field $\Q_p$ with a non-Archimedean norm $\norm {\,\cdot\,}_p$ such that:

$i) \quad$there exists a distance-preserving monomorphism $\phi:\struct{\Q,\norm {\,\cdot\,}_p} \to \struct{\Q_p,\norm {\,\cdot\,}_p}$.
$ii) \quad$the image $\map \phi \Q$ is dense in $\struct{\Q_p,\norm {\,\cdot\,}_p}$.
$iii) \quad \struct{\Q_p,\norm {\,\cdot\,}_p}$ is complete.

The valued field $\struct{\Q_p,\norm {\,\cdot\,}_p}$ is unique (up to isometric isomorphism).

Furthermore, by Normed Division Ring is Dense Subring of Completion the mapping $\phi: \Q \to \map \phi \Q$ is an isometric isomorphism onto the dense subfield $\map \phi \Q$ of $\Q_p$.

It is natural to identify $\Q$ with $\map \phi \Q$.

Also see


Sources