Editing Transcendental number theory (section)

==Transcendence==
{{Main|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 [[zero of a function|root]] in the [[complex number]]s.  That is, for any non-constant polynomial <math>P</math> with rational coefficients there will be a complex number <math>\alpha</math> such that <math>P(\alpha)=0</math>.  Transcendence theory is concerned with the converse question: given a complex number <math>\alpha</math>, is there a polynomial <math>P</math> with rational coefficients such that <math>P(\alpha)=0?</math> 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 {α<sub>1</sub>, α<sub>2</sub>, …, α<sub>''n''</sub>} is called algebraically independent over a [[field (mathematics)|field]] ''K'' if there is no non-zero polynomial ''P'' in ''n'' 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 ''n''&nbsp;=&thinsp;1 and the field ''K'' is the field of [[rational number]]s.

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.