Differential Galois theory
Whereas algebraic Galois theory studies extensions of algebraic fields, differential Galois theory studies extensions of differential fields, i.e. fields that are equipped with a derivation, D. Much of the theory of differential Galois theory is parallel to algebraic Galois theory. 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.
The problem of finding which integrals of elementary functions can be expressed with other elementary functions is analogous to the problem of solutions of polynomial equations by radicals in algebraic Galois theory, and is not addressed by differential Galois theory, but is solved by Liouville's theorem and the Risch algorithm.
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