  Differential Galois theory has played important roles in the theory of
integrability of linear differential equation. In this paper we will extend the
theory to nonlinear case and study the integrability of the first order
nonlinear differential equation. We will define for the differential equation
the differential Galois group, will study the structure of the group, and will
prove the equivalent between the existence of the Liouvillian first integral
and the solvability of the corresponding differential Galois group.
