Differential Galois theory

In mathematics, differential Galois theory is the field that studies extensions of differential fields.

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. A derivation satisfies the properties

D(xy) = (Dx)y + xDy

and

D(x + y) = Dx + Dy

for any field elements x and y. (These identities correspond to the Leibniz product rule and the linearity of derivatives) Much of the theory of differential Galois theory is analogous 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.

Most of differential Galois theory is analogous to algebraic Galois theory. 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.