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.
Suppose that is a formal power series with rational coefficients an, 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.
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 Funktionen (1852), reproduced in Mathematische Gesammelte Werke, Band II, Chelsea Publishing Co., New York, 1975, p. 765–767.
More recently, many authors have investigated precise and effective bounds quantifying the above almost all.