P-polynomials and H-Galois extensions

Abstract

Let R be a commutative ring with identity 1 and let H be a finite cyclic Hopf algebra, i.e., H is a Hopf algebra over R and has a free basis ( 1, D,..., DC’’ ) as R-module. A finite cyclic Hopf algebra H with free basis ( 1, D,..., D” ’ ) is called a derioation rtpe if the comultiplication map d of Hisdefined byd(D)=D@l+l@D: In this paper, for a finite cyclic Hopf algebra of derivation type, we determine the structure of H-Galois estrnsions of R in the sense of [9, Definition 1.41. (An H-Galois extension of R is equivalent to an H*-Hopf Galois extension of R in the sense of [ 17, p. 66, Definition], where H* is the dual Hopf algebra Hom.(H, R).) First we show that if H is a finite cyclic Hopf algebra of derivation type, then the base ring R contains the prime field GF(p) and the algebra structure of H is determined by a p-polynomial