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 Straßmann (1928).

Statement of the theorem[]

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 a_n 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 |a_N| = max |a_n|.

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), 159: 13–28, doi:10.1515/crll.1928.159.13, ISSN 0075-4102, JFM 54.0162.06

External links[]

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