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 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

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
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