Differential extension fields of exponential type.

The special properties of differential extension fields which can be generated by elements with logarithmic derivatives in the base field are worked out. The results are analogous to those for Kummer extensions of ordinary fields, where ntΆ roots are adjoined. The problem of the integration in finite terms of elements of such extension fields is considered, with applications to certain distribution functions that occur in statistics. 1* By a differential field is here meant a field k together with an indexed family {Di}ίeI of derivations of k. For brevity, we speak of "the differential field k", referring to the whole combination, and of "the given derivations of fc", referring to the set {Dt}ieI. The constants of the differential field k are f\ieIker Dif a subfield of k. A differential extension field of k is an extension field K of k together with a family of derivations {D'i}ieI of K indexed by the same set such that the restriction of each Ώ\ to k is A If fc is a differential field and x a nonzero element of some differential extension field K of k, we say that x is exponential over k if Dx/x e k for each given derivation D of K; in virtue of the "logarithmic derivative identity" Dxjx + Dy/y = D(xy)/(xy), the set of all elements of K that are exponential over k forms a multiplicative subgroup of K that contains the multiplicative group k* of k. Part of the following result occurs in [2, p. 1156]. THEOREM 1. Let k be a differential field of characteristi c zero, K a differential extension field of k with the same subfield of constants. Any element of K which can be written as a finite sum X^ yίf where each yt is an element of K that is exponential over k and yJVj gk if i Φ j, can be written as such a sum in only one way; in this case ^ yt is algebraic over k if and only if each yt is algebraic over kf which is true if and only if some power of y i is in k. If K = k(xJf •••,#*), with each xt exponential over k, then the multiplicative group E of all elements of K which are exponential over k is generated by x lf , xn and k*, and the abelian group Ejk* has rank deg. tr. K/k and torsion subgroup of order [(algebraic closure of k in K):k\.