On modular field extensions

Abstract

Let K be a purely inseparable extension of a field k of characteristic p, v’i , i = l)...) m be k-linear maps p’i : K -+ K, and t be a variable. Denoting by 1 the identity map of K, we call C#J~ = 1 + tvl + *a* + tmrpm an “approximate automorphism” of order m of K/k if G,(d) = (@,a)(@,b) mod tm+l for all a, b E K. If for every a E K with a $ k there is some approximate automorphism @, such that Qlta # a (or equivalently such that some yia # 0), then we shall say that K has “enough” approximate automorphisms. This is analogous to the requirement of normality for a finite separable extension, the latter having “enough” genuine automorphisms. It is reasonable to expect, therefore, that a purely inseparable extension with enough approximate automorphisms has further extraordinary properties, and in fact, Sweedler [2] has shown the following: