On r-primitive k-normal elements over finite fields

Let r, n be positive integers, k be a non-negative integer and q be any prime power such that r q(n) - 1. An element alpha of the finite field F-qn is called an r-primitive element, if its multiplicative order is (q(n) - 1)/r and it is called a k-normal element over F-q, if the degree of the greatest common divisor of the polynomials m alpha(x) = sigma(n)(i=1) alpha(qi-1) x(n-i )and xn( - 1) is k. In this article, we discuss the existence of an element alpha is an element of F-qn which is both r-primitive and k-normal over F-q. In particular, we show that there exists an element alpha E F-qn, which is both 2-primitive and 2-normal over F-q if and only if q is an odd prime power and either n >= 5 and gcd(q(3) - q, n) &NOTEQUexpressionL;1 or n = 4 and q equivalent to 1(mod 4).(C) 2022 Elsevier Inc. All rights reserved.