Editing Closed-form expression (section)

== Closed-form number ==
{{see also|Transcendental number theory}}
{{confusing|section|reason=as the section is written, it seems that Liouvillian numbers and elementary numbers are exactly the same|date=October 2020}}

Three subfields of the [[complex number]]s {{math|'''C'''}} have been suggested as encoding the notion of a "closed-form number"; in increasing order of generality, these are the Liouvillian numbers (not to be confused with [[Liouville number]]s in the sense of rational approximation), EL numbers and [[elementary number]]s. The '''Liouvillian numbers''', denoted {{math|'''L'''}}, form the smallest ''[[algebraically closed]]'' subfield of {{math|'''C'''}} closed under exponentiation and logarithm (formally, intersection of all such subfields)—that is, numbers which involve ''explicit'' exponentiation and logarithms, but allow explicit and ''implicit'' polynomials (roots of polynomials); this is defined in {{Harv|Ritt|1948|loc=p. 60}}. {{math|'''L'''}} was originally referred to as '''elementary numbers''', but this term is now used more broadly to refer to numbers defined explicitly or implicitly in terms of algebraic operations, exponentials, and logarithms. A narrower definition proposed in {{Harv|Chow|1999|loc=pp. 441–442}}, denoted {{math|'''E'''}}, and referred to as '''EL numbers''', is the smallest subfield of {{math|'''C'''}} closed under exponentiation and logarithm—this need not be algebraically closed, and corresponds to ''explicit'' algebraic, exponential, and logarithmic operations. "EL" stands both for "exponential–logarithmic" and as an abbreviation for "elementary".

Whether a number is a closed-form number is related to whether a number is [[transcendental number|transcendental]]. Formally, Liouvillian numbers and elementary numbers contain the [[algebraic number]]s, and they include some but not all transcendental numbers. In contrast, EL numbers do not contain all algebraic numbers, but do include some transcendental numbers. Closed-form numbers can be studied via [[transcendental number theory]], in which a major result is the [[Gelfond–Schneider theorem]], and a major open question is [[Schanuel's conjecture]].