Eisenstein's theorem
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In mathematics, Eisenstein's theorem is a result on the coefficients of any power series which is both an algebraic function, and with rational number coefficients. It applies easily to show that a function such as the exponential function must be a transcendental function. It was named after the German mathematician Ferdinand Eisenstein.
Suppose therefore 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. Let dn denote the denominator of an, as a fraction in lowest terms. Then Eisenstein's theorem states that there is a finite set S of prime numbers p, such that every prime factor of a number dn is contained in S.
This has an interpretation in terms of p-adic numbers: 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.
