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.

History

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

• p-adic exponential function

References

• Murty, M. Ram (2002). Introduction to P-Adic Analytic Number Theory. American Mathematical Society. p. 35. ISBN 978-0-8218-3262-2.
• Straßmann, Reinhold (1928), "Über den Wertevorrat von Potenzreihen im Gebiet der p-adischen Zahlen.", Journal für die reine und angewandte Mathematik (in German), 1928 (159): 13–28, doi:10.1515/crll.1928.159.13, ISSN 0075-4102, JFM 54.0162.06, S2CID 117410014

External links

• Weisstein, Eric W. "Strassman's Theorem". MathWorld.