A geometric characterization of the generators in an extension of odd prime order of a finite field (Q1068135)

If p is an odd prime, let \(K=GF(q)\) be the finite field of order \(q=p^ n\) and if \(r>1\) let \(K'=GF(q^ r)\) be the extension field of order \(q^ r\) so that K' is a vector space of dimension r over K. Let \(\Lambda\) and \(\Lambda\) ' be the sets of generators of the multiplicative groups \(K^{\times}\) and \((K')^{\times}\). In the \(r=2\) case \textit{R. E. Giudici} and \textit{C. Margaglio} [ibid. 62, 103-114 (1980; Zbl 0442.12018)] obtained geometric characterizations of \(\Lambda\) ' in terms of \(\Lambda\). In the present paper, the authors obtain analogous geometric results for the case where r is an odd prime.