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 $\Q_p$.

Each left coset $\sequence {x_n} + \NN \in \Q_p$ is called a $p$-adic number.

Also see

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