Strassmann's theorem

In mathematics, Strassmann's theorem is a result in field theory.

It states that, for suitable fields, suitable formal power series with coefficients in the valuation ring of the field have only finitely many zeroes.

Let K be a field with a non-Archimedean absolute value | · | and let R be the valuation ring of K. Let f(x) be a formal power series with coefficients in R other than the zero series, with coefficients an converging to zero with respect to | · |.

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 |aN| = max |an|.

As a corollary, there is no analogue of Euler's identity, e2πi = 1, in Cp, the field of p-adic complex numbers.