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