# Leigh.Samphier/Sandbox/Field Operations on P-adic Numbers

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## Theorem

Let $p$ be any prime number.

Let $\struct {\Q_p, \norm {\,\cdot\,}_p}$ be the $p$-adic numbers as quotient of Cauchy sequences.

Then the field operations on $\Q_p$ are defined by:

- $+ :\forall \eqclass{\sequence{x_n}}{}, \eqclass{\sequence{y_n}}{} \in \Q_p: \eqclass{\sequence{x_n}}{} + \eqclass{\sequence{y_n}}{} = \eqclass{\sequence{x_n + y_n}}{}$

- $\circ :\forall \eqclass{\sequence{x_n}}{}, \eqclass{\sequence{y_n}}{} \in \Q_p: \eqclass{\sequence{x_n}}{} \circ \eqclass{\sequence{y_n}}{} = \eqclass{\sequence{x_n y_n}}{}$

where $\eqclass{\sequence{x_n}}{}, \eqclass{\sequence{y_n}}{}$ denote the left coset of $\Q_p$ containing the Cauchy sequences $\sequence{x_n}, \sequence{y_n}$, respectively, of $\Q$.

## Proof

By definition of $p$-adic numbers as quotient of Cauchy sequences:

- $\Q_p$ is the quotient ring of Cauchy sequences of the valued field $\struct {\Q, \norm {\,\cdot\,}^\Q_p}$

By definition of the quotient ring of Cauchy sequences:

- $\Q_p = \CC / \NN$

where:

- $\CC$ is the ring of Cauchy sequences over $\Q$
- $\NN$ is the ideal of null sequences over $\Q$

From Leigh.Samphier/Sandbox/Addition of Quotient Ring of Cauchy Sequences:

- $+ :\forall \eqclass{\sequence{x_n}}{}, \eqclass{\sequence{y_n}}{} \in \Q_p: \eqclass{\sequence{x_n}}{} + \eqclass{\sequence{y_n}}{} = \eqclass{\sequence{x_n + y_n}}{}$

From Leigh.Samphier/Sandbox/Product of Quotient Ring of Cauchy Sequences:

- $\circ :\forall \eqclass{\sequence{x_n}}{}, \eqclass{\sequence{y_n}}{} \in \Q_p: \eqclass{\sequence{x_n}}{} \circ \eqclass{\sequence{y_n}}{} = \eqclass{\sequence{x_n y_n}}{}$

$\blacksquare$