Field Operations of P-adic Numbers as Quotient of Cauchy Sequences

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Theorem

Let $p$ be a prime number.

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


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

That is, $\Q_p$ is the quotient ring $\CC \, \big / \NN$ where:

$\CC$ denotes the commutative ring of Cauchy sequences over $\struct {\Q, \norm {\,\cdot\,}_p}$
$\NN$ denotes the set of null sequences in $\struct {\Q, \norm {\,\cdot\,}_p}$.


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

$+ :\quad \forall \sequence{x_n} + \NN, \sequence{y_n} + \NN \in \CC \, \big / \NN$:
$\quad \paren{\sequence{x_n} + \NN} + \paren{\sequence{y_n} + \NN} = \sequence{x_n + y_n} + \NN$
$\circ :\quad \forall \sequence{x_n} + \NN, \sequence{y_n} + \NN \in \CC \, \big / \NN$:
$\quad \paren{\sequence{x_n} + \NN} \paren{\sequence{y_n} + \NN} = \sequence{x_n y_n} + \NN$


Proof

By Corollary to Cauchy Sequences form Ring with Unity, $\CC$ is a commutative ring of Cauchy sequences with the ring operations defined by:

$+ :\quad \forall \sequence{x_n}, \sequence{y_n} \in \CC$:
$\quad \sequence{x_n} + \sequence{y_n} = \sequence{x_n + y_n}$
$\circ :\quad \forall \sequence{x_n}, \sequence{y_n} \in \CC$:
$\quad \sequence{x_n} \circ \sequence{y_n} = \sequence{x_n y_n}$


By Corollary to Null Sequences form Maximal Left and Right Ideal, $\NN$ is a maximal ideal of $\CC$.

By Corollary to Quotient Ring of Cauchy Sequences is Division Ring, the quotient ring $\Q_p = \CC \big / \NN$ is a field with field operations defined by:

$+ :\quad \forall \sequence{x_n} + \NN, \sequence{y_n} + \NN \in \CC \, \big / \NN$:
$\quad \paren{\sequence{x_n} + \NN} + \paren{\sequence{y_n} + \NN} = \paren{\sequence{x_n} + \sequence{y_n}} + \NN$
$\circ :\quad \forall \sequence{x_n} + \NN, \sequence{y_n} + \NN \in \CC \, \big / \NN$:
$\quad \paren{\sequence{x_n} + \NN} \paren{\sequence{y_n} + \NN} = \paren{\sequence{x_n} \circ \sequence{y_n}} + \NN$


Putting the operations on $\CC$ with those on $\CC \, \big / \NN$ gives:

$+ :\quad \forall \sequence{x_n} + \NN, \sequence{y_n} + \NN \in \CC \, \big / \NN$:
$\quad \paren{\sequence{x_n} + \NN} + \paren{\sequence{y_n} + \NN} = \sequence{x_n + y_n} + \NN$
$\circ :\quad \forall \sequence{x_n} + \NN, \sequence{y_n} + \NN \in \CC \, \big / \NN$:
$\quad \paren{\sequence{x_n} + \NN} \paren{\sequence{y_n} + \NN} = \sequence{x_n y_n} + \NN$

$\blacksquare$