Definition:P-adic Number/Representative
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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 quotitent ring of the ring of Cauchy sequences over $\struct {\Q, \norm {\,\cdot\,}_p}$ by null sequences in $\struct {\Q, \norm {\,\cdot\,}_p}$.
Let $\sequence{x_n}$ be Cauchy sequence in $\struct{\Q, \norm {\,\cdot\,}_p}$.
Let $\eqclass{x_n}{}$ denote the left coset of $\sequence{x_n}$ in $\Q_p$.
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
- Representative of P-adic Number is Representative of Equivalence Class where it is proved that the definition of a representative of a $p$-adic number coincides with the definition of a representative of an equivalence class.
Sources
- 2007: Svetlana Katok: p-adic Analysis Compared with Real: $\S 1.4$ The field of $p$-adic numbers $\Q_p$