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

This article studies the existence of elements α in finite fields 𝔽_q^n such that both α and its inverse α ^-1 are r -primitive and k -normal over 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 (α ,α ^-1) . Moreover, we find that for n≥ 7 , there always exists a pair (α ,α ^-1) of 1-primitive and 1-normal elements in 𝔽_q^n over 𝔽_q . Additionally, we obtain that for n=5,6 , if gcd(q,n)=1 , there always exists such a pair in 𝔽_q^n , except for the field 𝔽_4^5 .