Definition:P-adic Number/Quotient of Cauchy Sequences in P-adic Norm

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Definition

Let $p$ be any prime number.

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

By P-adic Norm is Non-Archimedean Norm then $\norm {\,\cdot\,}_p$ is a non-archimedean norm on $\Q$ and the pair $\struct {\Q, \norm {\,\cdot\,}_p}$ is a valued field.


Let $\CC$ be the commutative ring of Cauchy sequences over $\struct {\Q, \norm {\,\cdot\,}_p}$.

Let $\NN$ be the set of null sequences in $\struct {\Q, \norm {\,\cdot\,}_p}$.

Let $\Q_p$ denote the quotient ring $\CC \, \big / \NN$.

Let $\norm {\, \cdot \,}_p:\Q_p \to \R_{\ge 0}$ be the norm on the quotient ring $\Q_p$ defined by:

$\displaystyle \forall \sequence {x_n} + \NN: \norm {\sequence {x_n} + \NN }_p = \lim_{n \mathop \to \infty} \norm{x_n}_p$

By Corollary to Quotient Ring of Cauchy Sequences is Normed Division Ring, then $\struct {\Q_p, \norm {\, \cdot \,}_p}$ is a valued field.


Each left coset $\sequence {x_n} + \NN \in \CC \, \big / \NN$ is called a $p$-adic number.

The p-adic numbers is the valued field $\struct {\Q_p, \norm {\,\cdot\,}_p}$.


Representative of a $p$-adic Number

Each Cauchy sequence $\sequence {y_n}$ of the left coset $\sequence {x_n} + \NN$ is called a representative of the $p$-adic number $\sequence {x_n} + \NN$.


Field Operations on $p$-adic Numbers

By Field Operations of P-adic Numbers as Quotient of Cauchy Sequences, the field operations on $\Q_p$ are defined by:

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


Rational Numbers as Dense Subfield of $p$-adic Numbers

By Rational Numbers are Dense Subfield of P-adic Numbers, the mapping $\phi: \Q \to \Q_p$ defined by:

$\map \phi r = \sequence {r, r, r, \dotsc} + \NN$

where $\sequence {r, r, r, \dotsc} + \NN$ is the left coset in $\CC \, \big / \NN$ that contains the constant sequence $\sequence {r, r, r, \dotsc}$, defines an isometric isomorphism between $\Q$ and $\map \phi \Q$.

In addition, $\map \phi \Q$ is a dense subfield of $\Q_p$.

It is natural to identify $\Q$ with $\map \phi \Q$.


Also see

Sources