# P-adic Integers is Valuation Ring Induced by P-adic Norm/Corollary

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## Theorem

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

Then:

- $(a): \quad$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

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

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

From Valuation Ideal of P-adic Numbers:

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

From Corollary to Valuation Ideal is Maximal Ideal of Induced Valuation Ring:

- $(a):\quad \Z_p$ is a local ring
- $(b):\quad p\Z_p$ is the unique maximal ideal of $\Z_p$

$\blacksquare$

## Sources

- 1997: Fernando Q. Gouvea:
*p-adic Numbers: An Introduction*... (previous) ... (next): $\S 3.3$ Exploring $\Q_p$ - 2007: Svetlana Katok:
*p-adic Analysis Compared with Real*... (previous) ... (next): $\S 1.8$ Algebraic properties of $p$-adic integers: Corollary $1.45$