Definition:P-adic Norm/P-adic Numbers/Notation

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 $\norm {\,\cdot\,}^Q_p$ 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

 * Leigh.Samphier/Sandbox/Rational Numbers form 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 $\norm {\,\cdot\,}^{\Q}_p = \norm {\,\cdot\,}_p \restriction_\Q$