Transcendental number theory

Transcendental number theory is a branch of number theory that investigates transcendental numbers, in both qualitative and quantitative ways.

The fundamental theorem of algebra tells us that if we have a non-zero polynomial with integer coefficients then that polynomial will have a root in the complex numbers. That is, for any polynomial P with integer coefficients there will be a complex number α such that P(α) = 0. Transcendence theory is concerned with the converse question, given a complex number α, is there a polynomial P with integer coefficients such that P(α) = 0? 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,…,α_n} 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,…,α_n) = 0. So working out if a given number is transcendental is really a special case of algebraic independence where our set consists of just one number.

A related but broader notion than "algebraic" 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 log_ab was not algebraic for rational numbers a and b provided b is not of the form b = a^c for some rational c.

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