Definition:P-adic Number

Definition

Let $p$ be a prime number.

Let $\norm {\,\cdot\,}_p$ be the p-adic norm on the rationals $\Q$.

Let $\Q_p$ be the field of $p$-adic numbers.

That is, $\Q_p$ is the quotient ring of the ring of Cauchy sequences over $\struct {\Q, \norm {\,\cdot\,}_p}$ by null sequences in $\struct {\Q, \norm {\,\cdot\,}_p}$.

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 $\Q_p$.

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 $\eqclass{x_n}{}$.