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P-adic exponential function: Difference between revisions

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→Properties: correct ___domain on which log and exp are inverses
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Similarly if ''z'' and ''w'' are nonzero elements of '''C'''<sub>''p''</sub> then log<sub>''p''</sub>(''zw'')&nbsp;=&nbsp;log<sub>''p''</sub>''z''&nbsp;+&nbsp;log<sub>''p''</sub>''w''.
 
And for suitableFor ''z'', soin thatthe everything___domain isof definedexp<sub>''p''</sub>, we have exp<sub>''p''</sub>(log<sub>''p''</sub>(1+''z''))&nbsp;=&nbsp;1+''z'' and log<sub>''p''</sub>(exp<sub>''p''</sub>(''z''))&nbsp;=&nbsp;''z''.
 
The roots of the Iwasawa logarithm log<sub>''p''</sub>(''z'') are exactly the elements of '''C'''<sub>''p''</sub> of the form ''p<sup>r''</sup>·ζ where ''r'' is a rational number and ζ is a root of unity.<ref>{{harvnb|Cohen|2007|loc=Proposition 4.4.45}}</ref>
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