Definition:P-adic Number

Definition

Let $p$ be any prime number.

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

Let $\CC$ be the commutative ring of Cauchy sequences over $\struct {\Q, \norm {\,\cdot\,}_p}$.

Let $\NN$ be the set of null sequences in $\struct {\Q, \norm {\,\cdot\,}_p}$.

Let $\Q_p$ denote the quotient ring $\CC \, \big / \NN$.

Let $\norm {\, \cdot \,}_p:\Q_p \to \R_{\ge 0}$ be the norm on the quotient ring $\Q_p$ defined by:

$\ds \forall \sequence {x_n} + \NN: \norm {\sequence {x_n} + \NN }_p = \lim_{n \mathop \to \infty} \norm{x_n}_p$

Each left coset $\sequence {x_n} + \NN \in \CC \, \big / \NN$ is called a $p$-adic number.

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

Let $\sequence{x_n} + \NN$ be any $p$-adic number of $\Q_p$.

Each Cauchy sequence $\sequence {y_n}$ of the left coset $\sequence{x_n} + \NN$ is called a representative of the $p$-adic number $\sequence{x_n} + \NN$.

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

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 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.