Ideals of P-adic Integers/Corollary
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Theorem
Let $\Z_p$ be the $p$-adic integers for some prime $p$.
Then:
- $\Z_p$ is a principal ideal domain
Proof
From Ideals of P-adic Integers, all ideals of $\Z_p$ are the principal ideals:
- $\quad\set 0$
- $\quad\forall n \in \N : p^n \Z_p$
Hence $\Z_p$ is a principal ideal domain by definition.
$\blacksquare$
Sources
- 2007: Svetlana Katok: p-adic Analysis Compared with Real ... (previous) ... (next): $\S 1.8$ Algebraic properties of $p$-adic integers: Proposition $1.46$