Strassmann's theorem

It was introduced by Reinhold Strassman. ==Statement of the theorem==

Let K be a field with a non-Archimedean absolute value |\cdot| and let R be the valuation ring of K. Let f(x) = a_0 + a_1 x + a_2 x^2 + \dots be a formal power series which is not identically zero, with coefficients a_n \in R converging to zero with respect to |\cdot|. Then f(x) has only finitely many zeroes in R. More precisely, the number of zeros is at most N, where N is the largest index with |a_N| = \max_{n \geq 0} |a_n|. == Applications ==

A corollary of the theorem is that there is no analogue of Euler's identity, e^{2 \pi i} = 1 in \mathbb{C}_p, the field of p-adic complex numbers. Strassman's theorem may also be used to prove the Skolem-Mahler-Lech theorem, which states that the set of indices at which a linear recurrence sequence is equal to zero is composed of a union of finitely many arithmetic progressions and a finite set. == Related results ==

The Weierstrass preparation theorem over complete local rings generalises Strassman's theorem. While Strassman's theorem states that f(x) has at most N zeros in R, a corollary of the Weierstrass preparation theorem is that f(x) has exactly N zeros in the valuation ring of the algebraic closure of K. == See also ==