FINITE FIELD EXTENSIONS WITH THE LINE OR TRANSLATE PROPERTY FOR r-PRIMITIVE ELEMENTS

Let r; n > 1 be integers and q be any prime power q such that r vertical bar q(n) - 1. We say that the extension F-qn/F-q possesses the line property for r-primitive elements property if, for every alpha, theta is an element of F-qn(*) such that F-qn=F-q(theta), there exists some x is an element of F-q such that alpha(theta + x) has multiplicative order (q(n)-1)/r. We prove that, for sufficiently large prime powers q, F-qn/F-q possesses the line property for r-primitive elements. We also discuss the (weaker) translate property for extensions.