  Let $G$ be one of the classical groups of Lie rank $l$. We make a similar
construction of a general extension field in differential Galois theory for $G$
as E. Noether did in classical Galois theory for finite groups. More precisely,
we build a differential field $E$ of differential transcendence degree $l$ over
the constants on which the group $G$ acts and show that it is a Picard-Vessiot
extension of the field of invariants $E^G$. The field $E^G$ is differentially
generated by $l$ differential polynomials which are differentially
algebraically independent over the constants. They are the coefficients of the
defining equation of the extension. Finally we show that our construction
satisfies generic properties for a specific kind of $G$-primitive
Picard-Vessiot extensions.
