Generators for primary closures of Galois fields

We continue to study the existence of (norm- and) trace-compatible sequences of primitive normal bases for prime power extensions of finite fields, introduced by the author in Hachenberger (Finite Fields Appl. 5 (1999) 378–385; in: D. Jungnickel, H. Niederreiter (Eds.), Proceedings of the Fifth International Conference on Finite Fields and Applications, Augsburg, August 1999, Springer, Heidelberg, 2001, pp. 208–223), and improve on some aspects of these papers.