Inverses of r-primitive k-normal elements over finite fields

This article studies the existence of elements $$\alpha $$ in finite fields $$\mathbb {F}_{q^n}$$ such that both $$\alpha $$ and its inverse $$\alpha ^{-1}$$ are r-primitive and k-normal over $$\mathbb {F}_q$$ . We define a characteristic function for the set of k-normal elements and use it to establish a sufficient condition for the existence of the desired pair $$(\alpha ,\alpha ^{-1})$$ . Moreover, we find that for $$n\ge 7$$ , there always exists a pair $$(\alpha ,\alpha ^{-1})$$ of 1-primitive and 1-normal elements in $$\mathbb {F}_{q^n}$$ over $$\mathbb {F}_q$$ . Additionally, we obtain that for $$n=5,6$$ , if $$\textrm{gcd}(q,n)=1$$ , there always exists such a pair in $$\mathbb {F}_{q^n}$$ , except for the field $$\mathbb {F}_{4^5}$$ .