  We construct the field A generated by n algebraically independent elements,
and show that the linear space of derivations over this field is faithfully
represented by the linear space of the n-th fold Cartesian product of this
field acting through inner product on the gradient of this field. We prove also
that functional independence of a set in this field is equivalent to linear
independence of the gradient set in the space of Cartesian product. It is shown
that every subfield S of A which is generated by (n-1) functionally independent
elements defines an one-dimensional space of derivations, such that each member
L of the latter subspace has S as its kernel. Each coset of the multiplicative
subgroup S defines a non-homogeneous differential operator L+q whose kernel
coincide with this coset. We prove also that every element of A defines a coset
of the subgroup ker(L+q) in the additive group A, on which L+q is constant.
