On Primitive Elements in Differentially Algebraic Extension Fields

It is well known that if Fis a field of characteristic zero and K=F(ax,..., ot„) is a finite algebraic extension of F, then K contains a primitive element, i.e. an element a such that F(oti,..., an) = F(a). Moreover, by means of Galois theory, it is possible to characterize those elements of the extension field which are primitive. In the case of finite differentially algebraic extensions the theorem without further restrictions is false. Let Q be the field of rational numbers and 8 the usual derivation, i.e., 8q=0 for every q e Q. Let cx,.. .,cn be algebraically independent complex numbers over Q. If (Q(,cx,..., cn>, 8) is the differentially algebraic extension of Q where ac=0 for every c e Q(cx,..., cn>, then the underlying set of Q(cx,..., c„> is identical with that of Q(cx,..., cn), whence it is clear that there is no element c € Q(cx,..., cn> such that Q(cx,..., cn> = £?. Kolchin [2] (also [5, p. 52]) has shown the existence of primitive elements in the case where the differential field F(ax,..., an> has one derivation operator and the field F has an element/such that S/VOO). The differential fields (F(x0, ■ ■ -, xp}, D) considered in this paper are differentially algebraic over F, but F does not contain nonconstant elements. We prove the existence of primitive elements in the case where the derivation operator satisfies the conditions