Definition:Valued Field of P-adic Numbers

Definition
Let $p$ be any prime number.

Let $\Q_p$ be the field of $p$-adic numbers.

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

The p-adic numbers is the valued field $\struct {\Q_p, \norm {\,\cdot\,}_p}$.

Also see

 * P-adic Numbers form Non-Archimedean Valued Field for as proof that $\struct {\Q_p, \norm {\,\cdot\,}_p}$ is a valued field with a non-Archimedean norm.


 * P-adic Numbers form Completion of Rational Numbers with P-adic Norm for a proof that that $\struct {\Q_p, \norm {\,\cdot\,}_p}$ is the completion of $\struct {\Q, \norm {\,\cdot\,}_p}$ and is unique up to isometric isomorphism.


 * Rational Numbers are Dense Subfield of P-adic Numbers for a proof that $\struct {\Q, \norm {\,\cdot\,}_p}$ is isometrically isomorphic to a dense subfield of $\struct {\Q_p, \norm {\,\cdot\,}_p}$ and so $\Q$ can be identified with a dense subfield of the p-adic numbers.