Leigh.Samphier/Sandbox/Mapping Rational Numbers as Dense Subfield of P-adic Numbers

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Theorem

Let $p$ be any prime number.

Let $\struct {\Q, \norm {\,\cdot\,}^\Q_p}$ be the rational numbers with $p$-adic norm.

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

Let $d_p$ be the metric induced by $\norm {\, \cdot \,}_p$.

Let $\phi: \Q \to \Q_p$ be the mapping defined by:

$\map \phi r = \eqclass{\sequence{r, r, r, \dotsc}}{}$

where $\eqclass{\sequence{r, r, r, \dotsc}}{}$ is the left coset in $\Q_p$ that contains the constant sequence $\sequence{r, r, r, \dotsc}$.


Then the mapping $\phi: \Q \to \Q_p$ defines an isometric isomorphism between $\Q$ and $\map \phi \Q$.


In addition, $\map \phi \Q$ is a dense subfield of $\struct {\Q_p, d_p}$.


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 $\struct {\Q, \norm {\,\cdot\,}^\Q_p}$


From Leigh.Samphier/Sandbox/Embedding Division Ring into Quotient Ring of Cauchy Sequences:

$\phi$ is a distance-preserving ring monomorphism.

Hence $\phi$ is an isometric isomorphism between $\Q$ and $\map \phi \Q$.


From Leigh.Samphier/Sandbox/Quotient of Cauchy Sequences is Metric Completion:

$\map \phi \Q$ is a dense subset of $\struct {\Q_p, d_p}$

$\blacksquare$