Definition:P-adic Integer/Definition 2

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

An element $x \in \Q_p$ is called a $p$-adic integer the canonical expansion of $x$ contains only positive powers of $p$.

The set of all $p$-adic integers is usually denoted $\Z_p$.

Thus:
 * $\ds \Z_p = \set {\sum_{n \mathop = 0}^\infty d_n p^n : \forall n \in \N: 0 \le d_n < p} = \set{\ldots d_n \ldots d_3 d_2 d_1 d_0 : \forall n \in \N: 0 \le d_n < p}$

Also see

 * Leigh.Samphier/Sandbox/Equivalence of Definitions of P-adic Integer


 * Leigh.Samphier/Sandbox/P-adic Integers is Valuation Ring Induced by P-adic Norm


 * Corollary to Valuation Ideal is Maximal Ideal of Induced Valuation Ring for a proof that the set of $p$-adic integers $\Z_p$ is a local ring.