Higher Derivations and Field Extensions

Let K be a field having prime characteristic p. The following conditions on a subfield k of K are equivalent: (i) || K" (k) = k and K/k is separable, (ii) k is the field of constants of an infinite higher derivation defined in K. (iii) k is the field of constants of a set of infinite higher derivations defined in K. If K/k is separably generated and k is algebraically closed in K, then k is the field of constants of an infinite higher derivation in K. If K/k is finitely generated then k is the field of constants of an infinite higher derivation in K if and only if K/k is regular. Introduction. The relationship between field extensions and derivations was investigated by Baer [l ] in 1927. Baer obtained a characterization of those subfields k of the field K that are the fields of constants of derivations defined in K. In the prime characteristic case it was found that k is the field of constants of a nonzero derivation defined in K if and only if K/k is a purely inseparable extension having exponent one. Later Weisfield [7] generalized this result to finite higher derivations and purely inseparable extensions having higher exponent. The works of Weisfield [7] and Sweedler [6] yield the following: Let K be a field having prime characteristic. The following conditions on a subfield k of K are equivalent: (i) K/k is a purely inseparable modular extension with finite exponent, (ii) k is the field of constants of a finite higher derivation in K. (iii) k is the field of constants of a set of finite higher derivations in K. The purpose of this paper is to extend the above results to infinite higher derivations. The following is obtained: Let K be a field having prime characteristic p. The following conditions on a subfield k of K are equivalent: (i) K/k is separable and f\ Kp"ik) = k. (ii) k is the field of constants of an infinite higher derivation in K. (iii) k is the field of constants of a set of infinite higher derivations in K. A few comments should be made concerning the theory for characteristic zero fields. Baer [l] showed that in this case the subfields of K which are fields of constants of derivations in K are precisely those subfields algebraically closed in K. These subfields are also the fields of constants of the finite and infinite higher derivations in K. Received by the editors February 22, 1971. AMS (MOS) subject classifications (1970). Primary 12F10, 12F15.