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 field of $p$-adic numbers 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 $\Q_p$.
$p$-adic Number
Each left coset $\eqclass{x_n}{}$ in $\Q_p$ is called a $p$-adic number.
Representative of a $p$-adic Number
Each Cauchy sequence $\sequence {y_n}$ of the left coset $\eqclass{x_n}{}$ is called a representative of the $p$-adic number $\eqclass{x_n}{}$.
Also see
- Field Operations of $p$-adic Numbers for a proof that the field of $p$-adic numbers is a field and the field operations are defined by:
- $+ : \quad \forall \eqclass {x_n}{}, \eqclass {y_n}{} \in \Q_p : \quad \eqclass {x_n}{} + \eqclass {y_n}{} = \eqclass {x_n + y_n}{}$
- $\circ : \quad \forall \eqclass {x_n}{}, \eqclass {y_n}{} \in \Q_p : \quad \eqclass {x_n}{} \circ \eqclass {y_n}{} = \eqclass {x_n y_n}{}$