Editing Differential Galois theory (section)

== Basic properties ==
For a differential field ''F'', if ''G'' is a separable algebraic extension of ''F'', the derivation of ''F'' uniquely extends to a derivation of ''G''. Hence, ''G'' uniquely inherits the differential structure of ''F''.

Suppose ''F'' and ''G'' are differential fields satisfying Con(''F'') = Con(''G''), and ''G'' is an elementary differential extension of ''F''. Let ''a'' ∈ ''F'' and ''y'' ∈ ''G'' such that ''Dy'' = ''a'' (i.e., ''G'' contains the indefinite integral of ''a''). Then there exist ''c''<sub>1</sub>, …, ''c''<sub>''n''</sub> ∈ Con(''F'') and ''u''<sub>1</sub>, …, ''u''<sub>''n''</sub>, ''v'' ∈ ''F'' such that
: <math>a = c_1\frac{Du_1}{u_1} + \dotsb + c_n\frac{Du_n}{u_n} + Dv</math>
(Liouville's theorem). In other words, only functions whose indefinite integrals are elementary (i.e., at worst contained within the elementary differential extension of ''F'') have the form stated in the theorem. Intuitively, only elementary indefinite integrals can be expressed as the sum of a finite number of logarithms of simple functions.

If ''G''/''F'' is a Picard-Vessiot extension, then ''G'' being a Liouville extension of ''F'' is equivalent to the differential Galois group having a solvable identity component.<ref>
The connected component containing the identity of an algebraic group is called the identity component. It forms a normal subgroup.
</ref> Furthermore, ''G'' being a Liouville extension of ''F'' is equivalent to ''G'' being embeddable in some Liouville extension field of ''F''.