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.

It was introduced by Reinhold Strassman.¹

Let ${\displaystyle K}$ be a field with a non-Archimedean absolute value ${\displaystyle |\cdot |}$ and let ${\displaystyle R}$ be the valuation ring of ${\displaystyle K}$. Let ${\displaystyle f(x)=a_{0}+a_{1}x+a_{2}x^{2}+\dots }$ be a formal power series which is not identically zero, with coefficients ${\displaystyle a_{n}\in R}$ converging to zero with respect to ${\displaystyle |\cdot |}$. Then ${\displaystyle f(x)}$ has only finitely many zeroes in ${\displaystyle R}$. More precisely, the number of zeros is at most ${\displaystyle N}$, where ${\displaystyle N}$ is the largest index with ${\displaystyle |a_{N}|=\max _{n\geq 0}|a_{n}|}$.

A corollary of the theorem is that there is no analogue of Euler's identity, ${\displaystyle e^{2\pi i}=1}$ in ${\displaystyle \mathbb {C} _{p}}$, 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.

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