Editing P-adic number (section)

== Definition ==
There are several equivalent definitions of {{mvar|p}}-adic numbers. The one that is given here is relatively elementary, since it does not involve any other mathematical concepts than those introduced in the preceding sections. Other equivalent definitions use [[completion of a ring|completion]] of a [[discrete valuation ring]] (see {{slink||p-adic integers}}), [[completion of a metric space]] (see {{slink||Topological properties}}), or [[inverse limit]]s (see {{slink||Modular properties}}).

A {{mvar|p}}-adic number can be defined as a ''normalized {{mvar|p}}-adic series''. Since there are other equivalent definitions that are commonly used, one says often that a normalized {{mvar|p}}-adic series ''represents'' a {{mvar|p}}-adic number, instead of saying that it ''is'' a {{mvar|p}}-adic number.

One can say also that any {{mvar|p}}-adic series represents a {{mvar|p}}-adic number, since every {{mvar|p}}-adic series is equivalent to a unique normalized {{mvar|p}}-adic series. This is useful for defining operations (addition, subtraction, multiplication, division) of {{mvar|p}}-adic numbers: the result of such an operation is obtained by normalizing the result of the corresponding operation on series. This well defines operations on {{mvar|p}}-adic numbers, since the series operations are compatible with equivalence of {{mvar|p}}-adic series.

{{anchor|Field of p-adic numbers}}
With these operations, {{mvar|p}}-adic numbers form a [[field (mathematics)|field]] called the '''field of {{math|''p''}}-adic numbers''' and denoted <math>\Q_p</math> or <math>\mathbf Q_p.</math> There is a unique [[field homomorphism]] from the rational numbers into the {{mvar|p}}-adic numbers, which maps a rational number to its {{mvar|p}}-adic expansion. The [[image (mathematics)|image]] of this homomorphism is commonly identified with the field of rational numbers. This allows considering the {{math|''p''}}-adic numbers as an [[extension field]] of the rational numbers, and the rational numbers as a [[subfield (mathematics)|subfield]] of the {{math|''p''}}-adic numbers.

The ''valuation'' of a nonzero {{mvar|p}}-adic number {{mvar|x}}, commonly denoted <math>v_p(x),</math> is the exponent of {{mvar|p}} in the first nonzero term of every {{mvar|p}}-adic series that represents {{mvar|x}}. By convention, <math>v_p(0)=\infty;</math> that is, the valuation of zero is <math>\infty.</math> This valuation is a [[discrete valuation]]. The restriction of this valuation to the rational numbers is the {{mvar|p}}-adic valuation of <math>\Q,</math> that is, the exponent {{mvar|v}} in the factorization of a rational number as <math dosplay=inline≝>\tfrac nd p^v,</math> with both {{mvar|n}} and {{mvar|d}} [[coprime]] with {{mvar|p}}.