Primitive normal bases for towers of field extensions
Let \(q\) be a prime power, \(k,e\) positive integers, and \(F,K,E\) the finite fields of order \(q,q^k,q^{ek}\), respectively. Let \({\mathcal F}\) be the set of triples \((q,k,e)\) such that, for every \(a\in K\) which is normal over \(F\), there exists a primitive \(w_a\) in \(E\) which is normal over \(F\), and whose \((E,K)\)-trace equals \(a\). \textit{S. D. Cohen} and the author [Appl. Algebra Eng. Commun. Comput. 9, 383-403 (1999; Zbl 0931.12002)] proved that \((q,1,e)\in {\mathcal F}\) for all \(e\geq 2\) and all \(q\). In the paper under review the author considers the case \(k>1\) and proves the following results: 1. \((q,k,e)\in {\mathcal F}\) for all \(e\geq 2\) divisible by the characteristic of \(F\), all \(k\), and all \(q\). 2. There are at most finitely many triples \((q,k,e)\) with \(q\geq 17\), \(k\geq 1\), and \(e\geq 3\) with \((q,k,e)\not\in {\mathcal F}\). 3. \((q,k,e)\in {\mathcal F}\) for \(k\) and \(e\) powers of the same prime \(r\geq 5\) and \((q,2^{\alpha+3},2^\beta),(q,3^\alpha,3^\beta)\in {\mathcal F}\) for all \(\alpha\geq 0\) and \(\beta\geq 2\) for all \(q\). The last result implies the existence of trace-compatible sequences of primitive normal elements in towers of finite fields with constant extension degree \(r\), where \(r\geq 5\) is prime.
scientific article; zbMATH DE number 1369828