Editing P-adic number (section)

== ''p''-adic integers ==
The '''{{mvar|p}}-adic integers''' are the {{mvar|p}}-adic numbers with a nonnegative valuation.

A <math>p</math>-adic integer can be represented as a sequence
<math display="block"> x = (x_1 \operatorname{mod} p, ~ x_2 \operatorname{mod} p^2, ~ x_3 \operatorname{mod} p^3, ~ \ldots)</math>
of residues <math>x_e</math> mod <math>p^e</math> for each integer <math>e</math>, satisfying the compatibility relations <math>x_i \equiv x_j ~ (\operatorname{mod} p^i)</math> for <math>i < j</math>.

Every [[integer]] is a <math>p</math>-adic integer (including zero, since <math>0<\infty</math>). The rational numbers of the form <math display=inline> \tfrac nd p^k</math> with <math>d</math> coprime with <math>p</math> and <math>k\ge 0</math> are also <math>p</math>-adic integers (for the reason that <math>d</math> has an inverse mod <math>p^e</math> for every <math>e</math>).

The {{mvar|p}}-adic integers form a [[commutative ring]], denoted <math>\Z_p</math>  or <math>\mathbf Z_p</math>, that has the following properties.
* It is an [[integral domain]], since it is a [[subring]] of a field, or since the first term of the series representation of the product of two non zero {{mvar|p}}-adic series is the product of their first terms.
* The [[unit (ring theory)|units]] (invertible elements) of <math>\Z_p</math> are the {{mvar|p}}-adic numbers of valuation zero.
* It is a [[principal ideal domain]], such that each [[ideal (ring theory)|ideal]] is generated by a power of {{mvar|p}}.
* It is a [[local ring]] of [[Krull dimension]] one, since its only [[prime ideal]]s are the [[zero ideal]] and the ideal generated by {{mvar|p}}, the unique [[maximal ideal]].
* It is a [[discrete valuation ring]], since this results from the preceding properties.
* It is the [[completion of a ring|completion]] of the local ring <math>\Z_{(p)} = \{\tfrac nd \mid n, d \in \Z,\, d \not\in p\Z \},</math> which is the [[localization (commutative algebra)|localization]] of <math>\Z</math> at the prime ideal <math>p\Z.</math>

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