Rational Numbers are Dense Subfield of P-adic Numbers

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Theorem

Let $p$ be any prime number.

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

Let $\struct {\Q_p, \norm {\,\cdot\,}_p}$ be the $p$-adic numbers.

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

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

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


Then:

$\struct{\Q, \norm {\,\cdot\,}^{\Q}_p }$ is isometrically isomorphic to $\map \phi \Q$ which is a dense subfield of $\Q_p$.


That is, $\struct{\Q, \norm {\,\cdot\,}^{\Q}_p }$ can be identified as a dense subfield of $\struct {\Q_p, \norm {\,\cdot\,}_p}$ and $\norm {\,\cdot\,}_p$ as an extension of $\norm {\,\cdot\,}^\Q_p$.

Proof

From P-adic Numbers form Completion of Rational Numbers with P-adic Norm:

$\struct {\Q_p, \norm {\,\cdot\,}_p}$ is a completion of $\struct {\Q, \norm {\,\cdot\,}^\Q_p}$

From Embedding Division Ring into Quotient Ring of Cauchy Sequences:

the mapping $\phi: \Q \to \Q_p$ is a distance-preserving monomorphism.

From Normed Division Ring is Dense Subring of Completion:

$\struct {\Q, \norm {\, \cdot \,}^\Q_p }$ is isometrically isomorphic to $\struct {\map \phi \Q, \norm {\, \cdot \,}_p }$ which is a dense subfield of $\struct {\Q_p, \norm {\, \cdot \,}_p }$.

$\blacksquare$