Definition:Field of P-adic Numbers

Definition
Let $p$ be any prime number. Let $\norm {\,\cdot\,}_p^\Q$ be the p-adic norm on the rationals $\Q$.

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

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

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

The field of p-adic numbers is the field $\Q_p$.

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

Also see

 * Field Operations of P-adic Numbers for a proof that the $p$-adic number field is a field and the field operations are defined by:
 * $+ : \quad \forall \eqclass {x_n}{}, \eqclass {y_n}{} \in \Q_p : \quad \eqclass {x_n}{} + \eqclass {y_n}{} = \eqclass {x_n + y_n}{}$


 * $\circ : \quad \forall \eqclass {x_n}{}, \eqclass {y_n}{} \in \Q_p : \quad \eqclass {x_n}{} \circ \eqclass {y_n}{} = \eqclass {x_n y_n}{}$