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 $\Q_p$.

The $p$-adic numbers $\Q_p$ contains the rationals numbers $\Q$ (disregarding isomorphisms), and the $p$-adic norm on $\Q_p$ is an extension of the $p$-adic norm on $\Q$.

Notation
Since $\norm {\,\cdot\,}_p$ on $\Q_p$ is an extension of $\norm {\,\cdot\,}_p$ on $\Q$ there is generally no need to distinguish the the two norms as the context is usually sufficient to distinguish them. This is similar to the use of the absolute value $\size {\,\cdot\,}$ on the standard number classes.

Also See

 * Definition:P-adic Number for the definition of the $p$-adic numbers.


 * Definition:P-adic Norm for the definition of the $p$-adic norm on $\Q$


 * Completion Theorem for a proof that the completion of $\struct {\Q, \norm {\,\cdot\,}_p}$ exists and is unique up to isometric isomorphism.


 * Normed Division Ring is Dense Subring of Completion for a proof that $\struct {\Q, \norm {\,\cdot\,}_p}$ is isometrically isomorphic to a dense subfield of $\struct {\Q_p, \norm {\,\cdot\,}_p}$.