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