Primitive complete normal bases for regular extensions (Q2777594)

If \(q\) is a prime power and \(r\) is a prime, let ord\(_r (q)\) denote the multiplicative order of \(q\) modulo \(r\). The extension field \(E = \text{GF}(q^m)\) of the Galois field \(F = \text{GF}(q)\) is called regular if \(\text{gcd}(m, \operatorname {ord}_r(q)) = 1\) for all prime divisors \(r\) of \(m\) which are different from the characteristic \(p\) of \(F\). A primitive element \(w\) of \(E\) is called completely normal over \(F\) if \(w\) simultaneously generates a normal basis for \(E\) over every intermediate field \(K\) of \(E/F\). Determining if \(E\) necessarily contains such a completely normal primitive element is the basic problem under consideration in this paper. It is shown that indeed this is the case provided \(E\) is regular over \(F\) and also \(q \equiv 1\pmod 4\) whenever \(q\) is odd and \(m\) is even. The proof uses the theory of Gauss and character sums. The general case remains open and seems to be a very difficult problem.