P-adic Norm forms Non-Archimedean Valued Field/P-adic Numbers

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Theorem

Let $p$ be a prime number.

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

Then:

$\struct{\Q_p, \norm {\,\cdot\,}_p}$ is a valued field

$\norm {\,\cdot\,}_p$ is a non-Archimedean norm

That is, the $p$-adic numbers $\struct {\Q_p, \norm {\,\cdot\,}_p}$ form a valued field with a non-Archimedean norm.

Proof

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

From P-adic Norm on Rational Numbers is Non-Archimedean Norm:

$\struct{Q, \norm {\,\cdot\,}^\Q_p}$ is a valued field with non-Archimedean norm $\norm {\,\cdot\,}_p$

By definition of the $p$-adic numbers:

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

where:

$\CC$ is the commutative ring of Cauchy sequences over $\struct {\Q, \norm {\,\cdot\,}^\Q_p}$.

and

$\NN$ is the set of null sequences in $\struct {\Q, \norm {\,\cdot\,}^\Q_p}$.

From Corollary to Quotient Ring of Cauchy Sequences is Normed Division Ring:

$\struct {\Q_p, \norm {\, \cdot \,}_p}$ is a valued field.

From Completion of Normed Division Ring:

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

From Non-Archimedean Division Ring iff Non-Archimedean Completion:

$\norm {\, \cdot \,}_p$ on $\Q_p$ is a non-Archimedean norm.

$\blacksquare$