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.


$\blacksquare$

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