User tande - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-20T07:27:14Z http://mathoverflow.net/feeds/user/5668 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/22735/analytic-functions-over-fields-other-than-real-or-complex-numbers/22741#22741 Answer by tande for Analytic Functions over Fields other than Real or Complex Numbers tande 2010-04-27T16:29:54Z 2010-04-27T16:29:54Z <p>There is a perfectly working theory of analytic functions over the p-adics with lots of theorems. No model theory is needed (Neal Koblitz has a book about that, also Non-Archimedean Analysis by Bosch, Güntzer, Remmert for a dry treatise), but indeed we face (ultra)metric complete fields here, being uncountably infinite, just as you suggest.</p> <p>Nonetheless, if you just need the "feel" of power series to model something abstractly, formal power series, cf. <a href="http://en.wikipedia.org/wiki/Formal_power_series" rel="nofollow">http://en.wikipedia.org/wiki/Formal_power_series</a>, may be all you need. They behave in many ways like (convergent) power series, for example if you 'formally' wish to invert a differential operator, such computations - at least algebraically - may be given a more-or-less solid foundation in a formal power series ring.</p> <p>All classical operations, e.g. taking derivatives etc, can be defined termwise, no problem. You can also plug formal power series into each other, but just if the constant coefficient is zero, sadly.</p> <p>Finally, your two points do not really enforce large cardinality. A finite field can be equipped with the discrete metric, this makes it complete, so you could take about convergent power series over this - it just means that only finitely many coefficients can be non-zero, making it effectively a polynomial ring.</p>