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 (1990) 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