Pairs of r-Primitive and k-Normal Elements in Finite Fields

Let F q n be a finite field with q(n)-1 elements and r be a positive divisor of q(n)-1. An element a? F*q n is called r-primitive if its multiplicative order is(qn-1)/r. Also, a?F q nisk-normal over Fq if the greatest common divisor of the polynomials ga(x)=axn-1+aqxn-2+...+aqn-2x+aqn-1andxn-1inFqn[x]has de greek. These concepts generalize the ideas of primitive and normal elements, respectively. In this paper, we consider non-negative integersm1,m2,k1,k2, positive integersr1,r2andrational functions F(x)=F1(x)/F2(x)? F q n(x)with deg(Fi)= mifori ? {1,2}satisfying certain conditions and we present sufficient conditions for the existence ofr1-primitivek1-normal elementsa? Fq nover Fq, such that F(a)is anr2-primitivek2-normal element over F-q. Finally as an example we study the case wherer1=2,r2=3,k1=2,k2=1,m1=2 andm2=1, withn=7