# Leigh.Samphier/Sandbox/Rational Numbers with P-adic Norm is Non-Archimedean Valued Field

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## Theorem

Let $p$ be any prime number.

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

Then $\struct{\Q, \norm {\,\cdot\,}_p}$ is a non-Archimedean valued field.

## Proof

From P-adic Norm is Non-Archimedean Norm:

- $\norm {\,\cdot\,}_p$ is a non-Archimedean norm on the set of rational numbers.

Hence $\struct{\Q, \norm {\,\cdot\,}_p}$ is a non-Archimedean valued field by definition.

$\blacksquare$

## See also

- Definition of Non-Archimedean Division Ring Norm