linguistic:voice_change
p -adic integers

The p -adic integers are the p -adic numbers with a nonnegative valuation.

A p -adic integer can be represented as a sequence

of residues x mod p for each integer e, satisfying the compatibility relations x i ≡ x j (mod p i) {\displaystyle x_{i}\equiv x_{j}~(\operatorname {mod} p^{i})} for i < j.

Every integer is a p -adic integer (including zero, since 0 < ∞ {\displaystyle 0<\infty }). The rational numbers of the form n d p k {\textstyle {\tfrac {n}{d}}p^{k}} with d coprime with p and k ≥ 0 {\displaystyle k\geq 0} are also p -adic integers (for the reason that d has an inverse mod p for every e).

The p -adic integers form a commutative ring, denoted Z p {\displaystyle \mathbb {Z} _{p}} or Z p {\displaystyle \mathbf {Z} _{p}}, that has the following properties.

The last property provides a definition of the p -adic numbers that is equivalent to the above one: the field of the p -adic numbers is the field of fractions of the completion of the localization of the integers at the prime ideal generated by p.