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
[edit]It was introduced by Reinhold Straßmann (1928).
Statement of the theorem
[edit]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.
See also
[edit]References
[edit]- 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