Purely inseparable extensions and higher derivations
Morris Weisfeld · Transactions of the American Mathematical Society · 1965
I. Let F be a field having prime characteristic p and C be a subfield.An element x of F is called purely inseparable if x £ C and xpt £ C for some positive integer e; the least positive integer e such that x"e £ C is called the exponent of x over C. F is a purely inseparable extension of C if every element of F not in C is purely inseparable over C. The maximum of the set of exponents of the purely inseparable elements of F, if it exists, is called the exponent of F over C.In 1927 R. Baer studied the relationship between derivations and purely inseparable extensions having exponent one over the base field.In this paper a generalization of Baer's results to purely inseparable extensions having any exponent over the base field is studied.Of prime importance in this study is the notion due to H. Hasse and F. K. Schmidt of a higher derivation.Let A be a ring with an identity.The sequence D= {Z)(r)| 0 ^r<m\ of endomorphisms of (A, +), the additive group of A, is called a higher derivation in A if and only if Dl0) = I, the identity endomorphism of (A,+) and( 2is called a D-constant if and only if D(r)(x) = 0 for all r such that 0 < r < m.The set of Z)-constants of A is a subring which is closed with respect to taking multiplicative inverses.Let F be a purely inseparable extension of C. A subset B of F is called a sub-basis of F over C if and only if B fl C = 0, F = C(B), and, for any Received by the editors January 10, 1963.(') This paper originated in the author's doctoral dissertation, Yale University, New Haven, Conn., 1954.Cf. also A note on purely inseparable extensions, Bull.Amer.Math.Soc. 60 (1954), p. 336.The primary portions of this paper, except for revisions, were written when the author was a Fund for the Advancement of Education Teaching Intem at the University of Chicago.(2) Sil I denotes the sum of the elements of the indicated set.435