P-adic exponential function: Difference between revisions

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==''p''-adic logarithm function==

The power series

One can also define a '''''p''-adic logarithm function''' by the power series

:<math>\log_plog(1+zx)=\sum_{n=1}^\infty \frac{(-1)^{n+1}zx^n}{n},</math>

converges for ''x'' in '''C'''_''p'' satisfying |''x''|_''p''&nbsp;&lt;&nbsp;1 and so defines the '''''p''-adic logarithm function''' log_''p''(''z'') for |''z''&nbsp;&minus;&nbsp;1|_''p''&nbsp;&lt;&nbsp;1 satisfying the usual property log_''p''(''zw'')&nbsp;=&nbsp;log_''p''''z''&nbsp;+&nbsp;log_''p''''w''. The Butfunction thislog_''p'' can be extended to all of {{SubSup|'''C'''|''p''|×}} (the set of nonzero elements of '''C'''_''p'') by writingimposing anythat it continue to satisfy this last property and setting log_''zp''(''p'')&nbsp;=&nbsp;0. inSpecifically, every element ''w'' of {{SubSup|'''C'''<sub>|''p''|×}} can be written as ''w''&nbsp;=&nbsp;''p<sup>r''</subsup>·ζ·''z'' aswith ''r'' a rational number, ζ a root of unity, and |''z''&nbsp;=&minus;&nbsp;1|<sub>''p''<sup/sub>&nbsp;&lt;&nbsp;1,<ref>{{harvnb|Cohen|2007|loc=Proposition 4.4.44}}</ref> in which case log<sub>''mp''</supsub>·(''uw''·)&nbsp;=&nbsp;log_''vp'',(''z'').<ref>In wherefactoring ''mw'' as above, there is a rationalchoice number,of a root involved in writing ''up^r'' since ''r'' is rational; however, different choices differ only by multiplication by a root of unity, ofwhich ordergets coprimeabsorbed tointo the factor ζ.</ref> This function on {{SubSup|'''C'''|''p'',|×}} andis sometimes called the ''v'Iwasawa logarithm''' liesto inemphasize the original ___domainchoice of convergence for log_''p''(''p'')&nbsp;=&nbsp;0. In fact, sothere is an extension of the logarithm from |''vz''&nbsp;&minus;&nbsp;1|_''p''&nbsp;&lt;&nbsp;1. to Weall thenof define{{SubSup|'''C'''|''p''|×}} for each choice of log_''p''(''zp'') toin be log'''C'''_''p''(''v'').<ref>{{harvnb|Cohen|2007|loc=§4.4.11}}</ref>

==Properties==

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And for suitable ''z'', so that everything is defined, we have exp_''p''(log_''p''(''z''))&nbsp;=&nbsp;''z'' and log_''p''(exp_''p''(''z''))&nbsp;=&nbsp;''z''.

The roots of the Iwasawa logarithm log_''p''(''z'') are exactly the elements of '''C'''_''p'' of the form ''p^r''·ζ where ''r'' is a rational number and ζ is a root of unity.<ref>{{harvnb|Cohen|2007|loc=Proposition 4.4.45}}</ref>

Note that there is no analogue in '''C'''_''p'' of [[Euler's identity]], ''e''^2''πi''&nbsp;=&nbsp;1. This is a corollary of [[Strassmann's theorem]].

Another major difference to the situation in '''C''' is that the ___domain of convergence of exp_''p'' is much smaller than that of log_''p''. A modified exponential function &mdash; the [[Artin–Hasse exponential]] &mdash; can be used instead which converges on |''z''|_''p''&nbsp;&lt;&nbsp;1.

==Notes==

{{reflist}}

==References==

* Chapter 12 of {{cite book | last=Cassels | first=J. W. S. | authorlink=J. W. S. Cassels | title=Local fields | series=[[London Mathematical Society|London Mathematical Society Student Texts]] | publisher=[[Cambridge University Press]] | year=1986 | isbn=0-521-31525-5 }}

*{{Citation

| last=Cohen

| first=Henri

| author-link=Henri Cohen

| title=Number theory, Volume I: Tools and Diophantine equations

| publisher=Springer

| ___location=New York

| series=[[Graduate Texts in Mathematics]]

| volume=239

| year=2007

| isbn=978-0-387-49922-2

| id={{MR|2312337}}

| doi=10.1007/978-0-387-49923-9

}}

==External links==