Primitive normal bases with prescribed trace (Q1293965)

Let \(q\) be a prime power, \(F= GF(q)\) the Galois field of order \(q\) and \(E\) an extension field of degree \(n\) over \(F\). A nonzero element \(w\in E\) is called primitive if it generates the multiplicative group of \(E\). If the Frobenius orbit \(\{w^{q^i}\mid i= 0,\dots, n-1\}\subseteq E\) is a linearly independent subset, it forms a so-called normal basis of \(E\) over \(F\) and \(w\) is called an \(F\)-normal element. It is a classical result that normal bases always exist for \(E\). Due to a theorem of \textit{H. W. Lenstra} and \textit{R. J. Schoof} [Math. Comput. 48, 217-231 (1987; Zbl 0615.12023)], building on earlier work of \textit{L. Carlitz} [Trans. Am. Math. Soc. 73, 373-382 (1952; Zbl 0048.27302)] and \textit{H. Davenport} [J. Lond. Math. Soc. 43, 21-39 (1968; Zbl 0159.33901)], it is even known that there is always a primitive normal basis, i.e. a normal basis \(\{w^{q^i}\mid i= 0,\dots, n-1\}\) with \(w\in E\) a primitive element. In coding theory one is also interested in primitive elements \(w\in E\) with prescribed \(F\)-trace \(a:= \text{Tr}(w) = \sum_{i=0}^{n-1} w^{q^i}\in F\). These have been shown by S. D. Cohen to exist for arbitrary \(a\), \(n\geq 3\) and \((q,n)\neq (4,3)\) or for nonzero \(a\) in the remaining cases \((n\geq 2)\). In the present paper the authors combine these results of Cohen and of Lenstra and Schoof by proving the following theorem, which establishes a recent conjecture of \textit{I. H. Morgan} and \textit{G. L. Mullen} [Math. Comput. 63, 759-765, 519-523 (1994; Zbl 0805.11083)]. Theorem: Let \(E\) be a finite extension over a finite field \(F\) and let \(a\in F\) be nonzero. Then there exists an element \(w\in E\) which is primitive and \(F\)-normal and has \(F\)-trace equal to \(a\). The original conjecture of Morgan and Mullen was based on extensive computer searches for \(q\leq 97\) and \(n\leq 6\). It is therefore interesting to notice that, apart from two pairs \((F,E)\), the author's proof is purely theoretical.