Definition:P-adic Number

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.

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 $\struct {\Q, \norm {\,\cdot\,}_p}$