Eisenstein's theorem
Encyclopedia
In mathematics
, Eisenstein's theorem, named after the German mathematician Gotthold Eisenstein, applies to the coefficients of any power series which is an algebraic function
with rational number
coefficients. Through the theorem, it is readily demonstrable that a function such as the exponential function
must be a transcendental function
.
Suppose therefore that
is a formal power series
with rational coefficients an, which has a non-zero radius of convergence
in the complex plane
, and within it represents an analytic function
that is in fact an algebraic function. Let dn denote the denominator of an, as a fraction in lowest terms. Then Eisenstein's theorem states that there is a finite set S of prime number
s p, such that every prime factor of a number dn is contained in S.
This has an interpretation in terms of p-adic number
s: with an appropriate extension of the idea, the p-adic radius of convergence of the series is at least 1, for almost all
p (i.e. the primes outside the finite set S). In fact that statement is a little weaker, in that it disregards any initial partial sum of the series, in a way that may vary according to p. For the other primes the radius is non-zero.
Eisenstein's original paper is the short communication
Über eine allgemeine Eigenschaft der Reihen-Entwicklungen aller algebraischen Functionen
(1852), reproduced in Mathematische Gesammelte Werke, Band II, Chelsea Publishing Co., New York, 1975,
p. 765–767.
More recently, many authors have investigated precise and effective bounds quantifying the above almost all
.
See, e.g., Sections 11.4 and 11.55 of the book by E. Bombieri & W. Gubler.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, Eisenstein's theorem, named after the German mathematician Gotthold Eisenstein, applies to the coefficients of any power series which is an algebraic function
Algebraic function
In mathematics, an algebraic function is informally a function that satisfies a polynomial equation whose coefficients are themselves polynomials with rational coefficients. For example, an algebraic function in one variable x is a solution y for an equationwhere the coefficients ai are polynomial...
with rational number
Rational number
In mathematics, a rational number is any number that can be expressed as the quotient or fraction a/b of two integers, with the denominator b not equal to zero. Since b may be equal to 1, every integer is a rational number...
coefficients. Through the theorem, it is readily demonstrable that a function such as the exponential function
Exponential function
In mathematics, the exponential function is the function ex, where e is the number such that the function ex is its own derivative. The exponential function is used to model a relationship in which a constant change in the independent variable gives the same proportional change In mathematics,...
must be a transcendental function
Transcendental function
A transcendental function is a function that does not satisfy a polynomial equation whose coefficients are themselves polynomials, in contrast to an algebraic function, which does satisfy such an equation...
.
Suppose therefore that
is a formal power series
Formal power series
In mathematics, formal power series are a generalization of polynomials as formal objects, where the number of terms is allowed to be infinite; this implies giving up the possibility to substitute arbitrary values for indeterminates...
with rational coefficients an, which has a non-zero radius of convergence
Radius of convergence
In mathematics, the radius of convergence of a power series is a quantity, either a non-negative real number or ∞, that represents a domain in which the series will converge. Within the radius of convergence, a power series converges absolutely and uniformly on compacta as well...
in the complex plane
Complex plane
In mathematics, the complex plane or z-plane is a geometric representation of the complex numbers established by the real axis and the orthogonal imaginary axis...
, and within it represents an analytic function
Analytic function
In mathematics, an analytic function is a function that is locally given by a convergent power series. There exist both real analytic functions and complex analytic functions, categories that are similar in some ways, but different in others...
that is in fact an algebraic function. Let dn denote the denominator of an, as a fraction in lowest terms. Then Eisenstein's theorem states that there is a finite set S of prime number
Prime number
A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself. A natural number greater than 1 that is not a prime number is called a composite number. For example 5 is prime, as only 1 and 5 divide it, whereas 6 is composite, since it has the divisors 2...
s p, such that every prime factor of a number dn is contained in S.
This has an interpretation in terms of p-adic number
P-adic number
In mathematics, and chiefly number theory, the p-adic number system for any prime number p extends the ordinary arithmetic of the rational numbers in a way different from the extension of the rational number system to the real and complex number systems...
s: with an appropriate extension of the idea, the p-adic radius of convergence of the series is at least 1, for almost all
Almost all
In mathematics, the phrase "almost all" has a number of specialised uses."Almost all" is sometimes used synonymously with "all but finitely many" or "all but a countable set" ; see almost....
p (i.e. the primes outside the finite set S). In fact that statement is a little weaker, in that it disregards any initial partial sum of the series, in a way that may vary according to p. For the other primes the radius is non-zero.
Eisenstein's original paper is the short communication
Über eine allgemeine Eigenschaft der Reihen-Entwicklungen aller algebraischen Functionen
(1852), reproduced in Mathematische Gesammelte Werke, Band II, Chelsea Publishing Co., New York, 1975,
p. 765–767.
More recently, many authors have investigated precise and effective bounds quantifying the above almost all
Almost all
In mathematics, the phrase "almost all" has a number of specialised uses."Almost all" is sometimes used synonymously with "all but finitely many" or "all but a countable set" ; see almost....
.
See, e.g., Sections 11.4 and 11.55 of the book by E. Bombieri & W. Gubler.