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:

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

$\blacksquare$