Definition:P-adic Number

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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 $\eqclass{x_n}{}$.


Historical Note

The $p$-adic numbers were invented by Kurt Wilhelm Sebastian Hensel in $1897$ in order to bring the methods of power series to number theoretical problems.