Generators for primary closures of Galois fields.
Let \(q\) be a prime power, \(k\) and \(e\) positive integers and \((\mathbb F_q,\mathbb F_{q^k},\mathbb F_{q^{ek}})\) a tower of finite fields of order \(q\), \(q^k\) and \(q^{ek}\), respectively. Let \({\mathcal F}\) denote the set of triples \((q,k,e)\) for which, for every \(a\in \mathbb F_{q^k}\) normal over \(\mathbb F_q\) (i.e.\ \(\{a,a^q,\dots,a^{q^{k-1}}\}\) is a basis of \(\mathbb F_{q^k}\)), there exists a primitive element \(\gamma \in \mathbb F_{q^{ek}}\), normal over \(\mathbb F_q\), with \(\mathbb F_{q^k}\)-trace equal to \(a\). Theorem 1.1.: Let \(r\) be an odd prime different from the characteristic of \(\mathbb F_q\), \(s\) the order of \(q\) modulo \(r\) and assume that the order of \(q\) modulo \(r^2\) is equal to \(rs\). Then \((q,r^a,r^b)\in {\mathcal F}\) for all \(a\geq 0\) and \(b\geq 1\). This complements earlier results of the author [Finite Fields Appl. 5, No. 4, 378--385 (1999; Zbl 0982.11070)]. A quadruple \((q,k,l,n)\) with a prime power \(q\) and positive integers \(k,l,n\) such that \(k\) and \(l\) both divide \(n\) is called \textit{universal} if for any \(a\in \mathbb F_{q^k}\), normal over \(\mathbb F_q\), and any primitive element \(b\) of \(\mathbb F_{q^l}\), there exists a primitive element \(\gamma\) of \(\mathbb F_{q^n}\), normal over \(\mathbb F_q\), with \(\mathbb F_{q^k}\)-trace equal to \(a\) and \(\mathbb F_{q^l}\)-trace equal to \(b\). Theorem 1.2: Let \(r\geq 5\) be a prime different from the characteristic of \(\mathbb F_q\), \(s\) the order of \(q\) modulo \(r\) and assume that the order of \(q\) modulo \(r^2\) is equal to \(rs\). Then \((q,r^a,r^b,r^c)\) is a universal quadruple for all \(a,b\geq 0\) and all \(c>\max(a,b)\). This complements the author's earlier results in [Universal generators for primary closures of Galois fields, in Finite fields and applications. Proceedings of the fifth international conference on finite fields and applications \(F{_q^5}\), University of Augsburg, Germany, August 2--6, 1999, Berlin: Springer, 208--223 (2001; Zbl 1017.11066)].
completely normal (completely free) element
scientific article; zbMATH DE number 1891478