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

$(a): \quad$the $p$-adic integers, $\Z_p$, is a local ring

$(b): \quad$the principal ideal $p\Z_p$ is the unique maximal ideal of $\Z_p$

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 $\norm {\,\cdot\,}_p$

$\blacksquare$

Sources

• 1997: Fernando Q. Gouvea: p-adic Numbers: An Introduction ... (previous) ... (next): $\S 3.3$ Exploring $\Q_p$