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In [[mathematics]], '''Eisenstein's theorem''', named after the German mathematician [[Gotthold Eisenstein]], applies to the coefficients of any [[power series]] which is an [[algebraic function]] with [[rational number]] coefficients. Through the theorem, it is readily demonstrable that a function such as the [[exponential function]] must be a [[transcendental function]].  
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Suppose therefore that
 
:<math>\sum_{}^{} a_n t^n</math>
 
is a [[formal power series]] with rational coefficients ''a''<sub>''n''</sub>, which has a non-zero [[radius of convergence]] in the [[complex plane]], and within it represents an [[analytic function]] that is in fact an algebraic function. Let ''d''<sub>''n''</sub> denote the [[denominator]] of ''a''<sub>''n''</sub>, as a fraction [[in lowest terms]]. Then Eisenstein's theorem states that there is a finite set ''S'' of [[prime number]]s ''p'', such that every prime factor of a number ''d''<sub>''n''</sub> is contained in ''S''.
 
This has an interpretation in terms of [[p-adic number]]s: with an appropriate extension of the idea, the ''p''-adic radius of convergence of the series is at least 1, for [[almost all]] ''p'' (i.e. the primes outside the finite set ''S''). In fact that statement is a little weaker, in that it disregards any initial [[partial sum]] of the series, in a way that may ''vary'' according to ''p''. For the other primes the radius is non-zero.
 
Eisenstein's original paper is the short communication
''Über eine allgemeine Eigenschaft der Reihen-Entwicklungen aller algebraischen Functionen''
(1852), reproduced in Mathematische Gesammelte Werke, Band II, Chelsea Publishing Co., New York, 1975,
p.&nbsp;765–767.
 
More recently, many authors have investigated precise and effective bounds quantifying the above [[almost all]].
See, e.g.,  Sections 11.4 and 11.55 of the book by E. Bombieri & W. Gubler.
 
==References==
*{{Cite book|last = Bombieri|first = Enrico|authorlink=Enrico Bombieri|coauthors = [[Walter Gubler|Gubler, Walter]]|section=A local Eisenstein theorem|section=Power series, norms, and the local Eisenstein theorem|title=Heights in Diophantine Geometry|publisher=Cambridge University Press|year=2008|pages=362&ndash;376|doi= 10.2277/0521846153}}
 
{{DEFAULTSORT:Eisenstein's Theorem}}
[[Category:Theorems in number theory]]

Latest revision as of 23:33, 27 October 2014

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