Editing P-adic number (section)

== ''p''-adic series ==

The {{mvar|p}}-adic numbers are commonly defined by means of {{mvar|p}}-adic series.

A ''{{mvar|p}}-adic series'' is a [[formal power series]] of the form 
<math display="block">\sum_{i=v}^\infty r_i p^{i},</math>
where <math>v</math> is an integer and the <math>r_i</math> are rational numbers that either are zero or have a nonnegative valuation (that is, the denominator of <math>r_i</math> is not divisible by {{mvar|p}}).

Every rational number may be viewed as a {{mvar|p}}-adic series with a single nonzero term, consisting of its factorization of the form <math>p^k\tfrac nd,</math> with {{mvar|n}} and {{mvar|d}} both coprime with {{mvar|p}}.

Two {{mvar|p}}-adic series <math display=inline>\sum_{i=v}^\infty r_i p^{i} </math> and <math display=inline> \sum_{i=w}^\infty s_i p^{i} </math>
are ''equivalent'' if there is an integer {{mvar|N}} such that, for every integer <math>n>N,</math> the rational number 
<math display="block">\sum_{i=v}^n r_i p^{i} - \sum_{i=w}^n s_i p^{i} </math> 
is zero or has a {{mvar|p}}-adic valuation greater than {{mvar|n}}.

A {{mvar|p}}-adic series <math display=inline>\sum_{i=v}^\infty a_i p^{i} </math> is ''normalized'' if either all <math>a_i</math> are integers such that <math>0\le a_i <p,</math> and <math>a_v >0,</math> or all <math>a_i</math> are zero. In the latter case, the series is called the ''zero series''.

Every {{mvar|p}}-adic series is equivalent to exactly one normalized series. This normalized series is obtained by a sequence of transformations, which are equivalences of series; see [[#Normalization of a p-adic series|§ Normalization of a {{mvar|p}}-adic series]], below.

In other words, the equivalence of {{mvar|p}}-adic series is an [[equivalence relation]], and each [[equivalence class]] contains exactly one normalized {{mvar|p}}-adic series.

The usual operations of series (addition, subtraction, multiplication, division) are compatible with equivalence of {{mvar|p}}-adic series. That is, denoting the equivalence with {{math|~}}, if {{mvar|S}}, {{mvar|T}} and {{mvar|U}} are nonzero {{mvar|p}}-adic series such that <math>S\sim T,</math> one has
<math display="block">\begin{align}
S\pm U&\sim T\pm U,\\
SU&\sim TU,\\
1/S&\sim 1/T.
\end{align}</math>

The {{mvar|p}}-adic numbers are often defined as the equivalence classes of {{mvar|p}}-adic series, in a similar way as the definition of the real numbers as equivalence classes of [[Cauchy sequence]]s. The uniqueness property of normalization, allows uniquely representing any {{mvar|p}}-adic number by the corresponding normalized {{mvar|p}}-adic series. The compatibility of the series equivalence leads almost immediately to basic properties of {{mvar|p}}-adic numbers:
* ''Addition'', ''multiplication'' and [[multiplicative inverse]] of {{mvar|p}}-adic numbers are defined as for [[formal power series]], followed by the normalization of the result. 
* With these operations, the {{mvar|p}}-adic numbers form a [[field (mathematics)|field]], which is an [[extension field]] of the rational 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 non zero term of the corresponding normalized series; the valuation of zero is <math>v_p(0)=+\infty</math>
* The ''{{mvar|p}}-adic absolute value'' of a nonzero {{mvar|p}}-adic number {{mvar|x}}, is <math>|x|_p=p^{-v(x)};</math> for the zero {{mvar|p}}-adic number, one has <math>|0|_p=0.</math>

=== Normalization of a ''p''-adic series ===
Starting with the series <math display=inline>\sum_{i=v}^\infty r_i p^{i}, </math> the first above lemma allows getting an equivalent series such that the {{mvar|p}}-adic valuation of <math>r_v</math> is zero. For that, one considers the first nonzero <math>r_i.</math> If its {{mvar|p}}-adic valuation is zero, it suffices to change {{mvar|v}} into {{mvar|i}}, that is to start the summation from {{mvar|v}}. Otherwise, the {{mvar|p}}-adic valuation of <math>r_i</math> is <math>j>0,</math> and <math>r_i= p^js_i</math> where the valuation of <math>s_i</math> is zero; so, one gets an equivalent series by changing <math>r_i</math> to {{math|0}} and <math>r_{i+j}</math> to <math>r_{i+j} + s_i.</math> Iterating this process, one gets eventually, possibly after infinitely many steps, an equivalent series that either is the zero series or is a series such that the valuation of <math>r_v</math> is zero.

Then, if the series is not normalized, consider the first nonzero <math>r_i</math> that is not an integer in the interval <math>[0,p-1].</math> The second above lemma allows writing it <math>r_i=a_i+ps_i;</math> one gets n equivalent series by replacing <math>r_i</math> with <math>a_i,</math> and adding <math>s_i</math> to <math>r_{i+1}.</math> Iterating this process, possibly infinitely many times, provides eventually the desired normalized {{math|p}}-adic series.