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Concept
Transcendental number theory
Summary
Transcendental number theory is a branch of number theory that investigates transcendental numbers (numbers that are not solutions of any polynomial equation with rational coefficients), in both qualitative and quantitative ways.
Transcendental number
The fundamental theorem of algebra tells us that if we have a non-constant polynomial with rational coefficients (or equivalently, by clearing denominators, with integer coefficients) then that polynomial will have a root in the complex numbers. That is, for any non-constant polynomial with rational coefficients there will be a complex number such that . Transcendence theory is concerned with the converse question: given a complex number , is there a polynomial with rational coefficients such that If no such polynomial exists then the number is called transcendental.
More generally the theory deals with algebraic independence of numbers. A set of numbers {α1, α2, ..., αn} is called algebraically independent over a field K if there is no non-zero polynomial P in n variables with coefficients in K such that P(α1, α2, ..., αn) = 0. So working out if a given number is transcendental is really a special case of algebraic independence where n = 1 and the field K is the field of rational numbers.
A related notion is whether there is a closed-form expression for a number, including exponentials and logarithms as well as algebraic operations. There are various definitions of "closed-form", and questions about closed-form can often be reduced to questions about transcendence.
Use of the term transcendental to refer to an object that is not algebraic dates back to the seventeenth century, when Gottfried Leibniz proved that the sine function was not an algebraic function. The question of whether certain classes of numbers could be transcendental dates back to 1748 when Euler asserted that the number logab was not algebraic for rational numbers a and b provided b is not of the form b = ac for some rational c.
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