Editing Differential Galois theory (section)

== Definition ==
For any differential field ''F'' with derivation ''D'', there exists a subfield called the '''field of constants''' of ''F'', defined as:
: Con(''F'') = {''f'' ∈ ''F'' | ''Df'' = 0}.
The field of constants contains the prime field of ''F''.

Given two differential fields ''F'' and ''G'', ''G'' is called a simple differential extension of ''F'' if
<ref>
''G'' is a simple differential extension of ''F'' if for some element ''t'', ''G'' = ''F''<''t''> := ''F''(''t'', ''Dt'', ''D''(''Dt''), …).
</ref>
and satisfies
: ∃''s''∈''F''; ''Dt'' = ''Ds''/''s'',
then ''G'' is called a '''logarithmic extension''' of ''F''.

This has the form of a logarithmic derivative. Intuitively, ''t'' can be thought of as the logarithm of some element ''s'' in ''F'', corresponding to the usual chain rule. ''F'' does not necessarily have a uniquely defined logarithm. Various logarithmic extensions of ''F'' can be considered. Similarly, an '''exponential extension'''  satisfies
: ∃''s''∈''F''; ''Dt'' = ''tDs'',
and a '''differential extension''' satisfies
: ∃''s''∈''F''; ''Dt'' = ''s''.
A differential extension or exponential extension becomes a Picard-Vessiot extension when the field has characteristic zero and the constant fields of the extended fields match.

Keeping the above caveat in mind, this element can be regarded as the exponential of an element ''s'' in ''F''. Finally, if there is a finite sequence of intermediate fields from ''F'' to ''G'' with Con(''F'') = Con(''G''), such that each extension in the sequence is either a finite algebraic extension, a logarithmic extension, or an exponential extension, then ''G'' is called an '''elementary differential extension''' .

Consider the homogeneous [[linear differential equation]] for <math>a_1, \cdots , a_n \in F</math>:
: <math>D^{n}y + a_{1}D^{n-1}y + \cdots + a_{n-1}Dy + a_{n}y = 0</math>　…　(1).
There exist at most ''n'' linearly independent solutions over the field of constants. An extension ''G'' of ''F'' is a '''Picard-Vessiot extension''' for the differential equation (1) if ''G'' is generated by all solutions of (1) and satisfies Con(''F'') = Con(''G'').

An extension ''G'' of ''F'' is a '''Liouville extension'''  if Con(''F'') = Con(''G'') is an algebraically closed field, and there exists an increasing chain of subfields
: ''F'' = ''F''<sub>0</sub> ⊂ ''F''<sub>1</sub> ⊂ … ⊂ ''F''<sub>n</sub> = ''G''
such that each extension ''F''<sub>''k''+1</sub> : ''F<sub>k</sub>'' is either a finite algebraic extension, a differential extension, or an exponential extension. A Liouville extension of the rational function field '''C'''(''x'') consists of functions obtained by finite combinations of rational functions, exponential functions, roots of algebraic equations, and their indefinite integrals. Clearly, logarithmic functions, trigonometric functions, and their inverses are Liouville functions over '''C'''(''x''), and especially elementary differential extensions are Liouville extensions.

An example of a function that is contained in an elementary extension over '''C'''(''x'') but not in a Liouville extension is the indefinite integral of <math>e^{-x^2}</math>.