Definition:P-adic Norm/P-adic Numbers

Definition
Let $p$ be a prime number.

Let $\struct {\Q_p, \norm {\,\cdot\,}_p}$ be the $p$-adic numbers.

The norm $\norm {\,\cdot\,}_p$ on $\Q_p$ is called the $p$-adic norm on $\Q_p$.

Also see

 * Rational Numbers are Dense Subfield of P-adic Numbers for a proof that the $p$-adic norm on $p$-adic numbers may be considered an extension of the $p$-adic norm on the rational numbers, which means we may say $\norm {\,\cdot\,}^{\Q}_p = \norm {\,\cdot\,}_p \restriction_\Q$