Definition:P-adic Norm/P-adic Numbers

Definition
Let $p$ be a prime number.

Let $\norm {\,\cdot\,}_p$ denote the p-adic norm on the rationals $\Q$.

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

That is, $\Q_p$ is the quotitent ring of the ring of Cauchy sequences over $\struct {\Q, \norm {\,\cdot\,}_p}$ by null sequences in $\struct {\Q, \norm {\,\cdot\,}_p}$.

For any Cauchy sequence $\sequence{x_n}$ in $\struct{\Q, \norm {\,\cdot\,}_p}$, let $\eqclass{x_n}{}$ denote the left coset of $\sequence{x_n}$ in $\Q_p$.

Let $\norm {\, \cdot \,}_p:\Q_p \to \R_{\ge 0}$ be the norm on the quotient ring $\Q_p$ defined by:
 * $\ds \forall \eqclass{x_n}{} \in \Q_p: \norm {\eqclass{x_n}{} }_p = \lim_{n \mathop \to \infty} \norm{x_n}_p$

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

Also see

 * P-adic Norm is Non-Archimedean Norm for a proof that $\norm {\,\cdot\,}_p$ is a non-archimedean norm on $\Q$ and the pair $\struct {\Q, \norm {\,\cdot\,}_p}$ is a valued field.


 * P-adic Norm not Complete on Rational Numbers for a proof that $\struct {\Q, \norm {\,\cdot\,}_p}$ is not a complete valued field.


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

P-adic Numbers|$p$-adic numbers]] may be considered an extension of the $p$-adic norm on the rational numbers.
 * Rational Numbers are Dense Subfield of P-adic Numbers for a proof that the $p$-adic norm on Definition:* [[Definition:Valued Field of P-adic Numbers