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 $\struct {\Q_p, \norm {\,\cdot\,}_p}$.

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 $\struct {\Q_p, \norm {\,\cdot\,}_p}$ 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.


 * Field Operations of P-adic Numbers as Quotient of Cauchy Sequences