P-adic Integers is Valuation Ring Induced by P-adic Norm

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Theorem

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


Then:

the $p$-adic integers, $\Z_p$, is the valuation ring induced by the non-Archimedean norm $\norm {\,\cdot\,}_p$


Corollary

the $p$-adic integers, $\Z_p$, is a local ring


Proof

By the definition of the $p$-adic integers:

$\Z_p = \set {x \in \Q_p : \norm x_p \le 1}$

From P-adic Numbers form Non-Archimedean Valued Field:

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

By definition of the valuation ring induced by a non-Archimedean norm:

$\Z_p$ is the valuation ring induced by $\norm {\,\cdot\,}_p$

$\blacksquare$

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