  We study a necessary condition for the integrability of the polynomials
fields in the plane by means of the differential Galois theory. More
concretely, by means of the variational equations around a particular solution
it is obtained a necessary condition for the existence of a rational first
integral. The method is systematic starting with the first order variational
equation. We illustrate this result with several families of examples. A key
point is to check wether a suitable primitive is elementary or not. Using a
theorem by Liouville, the problem is equivalent to the existence of a rational
solution of a certain first order linear equation, the Risch equation. This is
a classical problem studied by Risch in 1969, and the solution is given by the
"Risch algorithm". In this way we point out the connection of the non
integrablity with some higher transcendent functions, like the error function.
