Definition:Field of P-adic Numbers

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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

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