Definition:Field of P-adic Numbers

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 $\CC \, \big / \NN$.

The p-adic number field is the field $\Q_p$.

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 $\Q_p$.

Also see

 * User:Leigh.Samphier/P-adic Number Field is Field
 * Field Operations of P-adic Numbers