Definition:P-adic Integer

Definition

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

Definition 1

An element $x \in \Q_p$ is called a $p$-adic integer if and only if $\norm x_p \le 1$.

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

Thus:

$\Z_p = \set {x \in \Q_p: \norm x_p \le 1}$

Definition 2

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

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

Thus:

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

Notation

The notation $\Z_p$ is also used for the ring of integers modulo $p$ where $p$ is a prime number.

On $\mathsf{Pr} \infty \mathsf{fWiki}$ the context of any page where $\Z_p$ appears will define what is referred to by $\Z_p$.

Also see

• Equivalence of Definitions of P-adic Integer

• P-adic Integers is Valuation Ring Induced by P-adic Norm

• P-adic Integers is Local Ring for a proof that the set of $p$-adic integers $\Z_p$ is a local ring.