Transcendence Theory - Transcendence

Transcendence

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.