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 is called algebraically independent over a field k
if there is no non-zero polynomial P
variables with coefficients in k
such that P
(α<sub>1</sub>,α<sub>2</sub>,…,α<sub>n</sub>) = 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.
Approximation by rational......