Editing P-adic number (section)

=== 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.