In
mathematics, Strassmann's theorem is a result in
field theory. It states that, for suitable fields, suitable formal
power series
In mathematics, a power series (in one variable) is an infinite series of the form
\sum_^\infty a_n \left(x - c\right)^n = a_0 + a_1 (x - c) + a_2 (x - c)^2 + \dots
where ''an'' represents the coefficient of the ''n''th term and ''c'' is a con ...
with coefficients in the
valuation ring In abstract algebra, a valuation ring is an integral domain ''D'' such that for every element ''x'' of its field of fractions ''F'', at least one of ''x'' or ''x''−1 belongs to ''D''.
Given a field ''F'', if ''D'' is a subring of ''F'' such ...
of the field have only finitely many zeroes.
History
It was introduced by .
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'', .
As a corollary, there is no analogue of
Euler's identity
In mathematics, Euler's identity (also known as Euler's equation) is the equality
e^ + 1 = 0
where
: is Euler's number, the base of natural logarithms,
: is the imaginary unit, which by definition satisfies , and
: is pi, the ratio of the circ ...
, ''e''
2''πi'' = 1, in C
''p'', the field of
p-adic
In mathematics, the -adic number system for any prime number extends the ordinary arithmetic of the rational numbers in a different way from the extension of the rational number system to the real and complex number systems. The extension ...
complex numbers.
See also
*
''p''-adic exponential function
References
*
*
External links
* {{MathWorld , title=Strassman's Theorem , urlname=StrassmansTheorem
Field (mathematics)
Theorems in abstract algebra