Pencils of Higher Derivations of Arbitrary Field Extensions

Let L be a field of characteristic p ^ 0. A subfield K of L is Galois if A' is the field of constants of a group of pencils of higher derivations on L. Let F d K be Galois subfields of L. Then the group of L over F is a normal subgroup of the group of L over K if and only if F = K(W') for some nonnegative integer r. If L/K splits as the tensor product of a purely inseparable extension and a separable extension, then the algebraic closure of X_in L, K, is also Galois in L. Given K, for every Galois extension Loi K, K is also Galois in L if and only if [K : Kp] < oo. 0. Introduction. Throughout we assume L is a field of characteristic p =£ 0. A rank / higher derivation on L is a sequence d = {d¡\0 < / < t + 1} of additive maps of L into L such that dr(ab)=^{dl(a)dJ(b)\i+j = r) and d0 is the identity map. The set of all rank / higher derivations forms a group with respect to the composition d ° e = /where fj = ~2{dmen\m + n = j}. Let H (L/K) be the set of all higher derivations on L trivial on K and having rank some power of p. Given d in H (L/K), v(d) = /where rank/ = p(rank d), f ¡ = d¡ and jÇ = 0 if p \j. Two higher derivations / and g are equivalent if g = v'(f) or/ = v'(g) for some i. The equivalence class of d is J and is called the pencil of d. The set of all pencils, H (L/K), can be given a group structure by defining df to be the pencil of d'f where d'Ed,f'Ef and rank d' = rank/' [3]. A subfield K of L will be called Galois if K is the field of constants of a group of pencils on L or equivalently if L/K is modular and n,AT(Z/') = K [2, Proposition 1]. In §1 it is shown that if F D K are Galois subfields of L, then H(L/F) is an invariant subgroup of H(L/K) if and only if F = K(LP) for some nonnegative integer r. This generalizes the result given in [2, Theorem 8] for the bounded exponent finite transcendence degree case. Let K denote the algebraic closure of K in L. L/K is said to split when L = J ®K D where J/K is purely inseparable and D/K is separable. §2 examines the question of when K is Galois in L, given L/K is Galois. Sufficient conditions are shown to be the splitting of L/K. Moreover, for every Galois extension L of K, K is also Galois in L if and only if Presented to the Society, August 10, 1978 under the title On pencil Galois theory; received by the editors March 8, 1978. AMS (MOS) subject classifications (1970). Primary 12F15.