Editing P-adic number (section)

== Basic lemmas ==

The theory of {{mvar|p}}-adic numbers is fundamentally based on the two following lemmas:

''Every nonzero rational number can be written <math display=inline>p^v\frac{m}{n},</math> where {{mvar|v}}, {{mvar|m}}, and {{mvar|n}} are integers and neither {{mvar|m}} nor {{mvar|n}} is divisible by {{mvar|p}}.'' The exponent {{mvar|v}} is uniquely determined by the rational number and is called its ''{{mvar|p}}-adic valuation'' (this definition is a particular case of a more general definition, given below). The proof of the lemma results directly from the [[fundamental theorem of arithmetic]].

''Every nonzero rational number {{mvar|r}} of valuation {{mvar|v}} can be uniquely written <math>r=ap^v+ s,</math> where {{mvar|s}} is a rational number of valuation greater than {{mvar|v}}, and {{mvar|a}} is an integer such that <math>0<a<p.</math>''

The proof of this lemma results from [[modular arithmetic]]: By the above lemma, <math display=inline>r=p^v\frac{m}{n},</math> where {{mvar|m}} and {{mvar|n}} are integers [[coprime]] with {{mvar|p}}. 
By [[Bézout's lemma]], there exist integers {{mvar|a}} and {{mvar|b}}, with <math>0\leq a < p</math>, such that 
<math> m  =  a n + b p.</math> Setting <math> s = b/n</math> (hence <math>{\rm val}(s) \geq 0</math>), we have
<math display="block"> {m\over n} = a + p {b \over n},\quad {\rm or} \quad r = a p^v  + p^{v + 1} s.</math>

To show the uniqueness of this representation, observe that if <math> r = a' p^v + p^{v + 1} s',</math> with 
<math>0\leq a' < p</math> and <math>{\rm val}(s')\geq 0</math>,
there holds by difference <math>(a -a') + p(s- s') = 0,</math> with <math>|a - a'| < p</math> and <math>{\rm val}(s-s') \geq 0</math>.
Write <math> s-s' = c/d</math>, where {{mvar|d}} is coprime to {{mvar|p}}; then 
<math>(a - a')d + p c = 0</math>, which is possible only if <math>a - a' = 0</math> and <math>c=0</math>.
Hence <math>a = a'</math> and <math> s = s'</math>.   

The above process can be iterated starting from {{mvar|s}} instead of {{mvar|r}}, giving the following.

''Given a nonzero rational number {{mvar|r}} of valuation {{mvar|v}} and a positive integer {{mvar|k}}, there are a rational number <math>s_k</math> of nonnegative valuation and {{mvar|k}} uniquely  defined nonnegative integers <math>a_0, \ldots, a_{k-1}</math> less than {{mvar|p}} such that <math>a_0>0</math> and''
<math display="block">r=a_0p^v + a_1 p^{v+1} +\cdots + a_{k-1}p^{v+k-1} +p^{v+k}s_k.</math>

The {{mvar|p}}-adic numbers are essentially obtained by continuing this infinitely to produce an [[infinite series]].