One difference between the two constructions is that the Galois groups in differential Galois theory tend to be matrix Lie groups, as compared with the finite groups often encountered in algebraic Galois theory.
It's important to note that the concept of elementary functions is merely conventional.
Using the theory of differential Galois theory , it is possible to determine which indefinite integrals of elementary functions cannot be expressed as elementary functions.
The significant difference in the structure is that the Galois group in differential Galois theory is an algebraic group, whereas in algebraic Galois theory, it is a profinite group equipped with the Krull topology.
Intuitively, t can be thought of as the logarithm of some element s in F, corresponding to the usual chain rule.
Consider the homogeneous linear differential equation for
: There exist at most n linearly independent solutions over the field of constants.
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 c1, …, cn ∈ Con(F) and u1, …, un, v ∈ F such that (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.
[2] Furthermore, G being a Liouville extension of F is equivalent to G being embeddable in some Liouville extension field of F. Differential Galois theory has numerous applications in mathematics and physics.
It is used, for instance, in determining whether a given differential equation can be solved by quadrature (integration).
One significant application is the analysis of integrability conditions for differential equations, which has implications in the study of symmetries and conservation laws in physics.