An Algebraic Treatment of Totally Linear Partial Differential Equations

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.