Completely normal primitive basis generators of finite fields

Ilene H. Morgan, Gary L. Mullen

Research output: Contribution to journalArticlepeer-review

7 Scopus citations


For q a prime power and n ≥ 2 an integer we consider the existence of completely normal primitive elements in finite field Fqn. Such an element α ∈ Fqn simultaneously generates a normal basis of Fqn over all subfields Fqd where d divides n. In addition, α multiplicatively generates the group of all nonzero elements of Fqn. For each pn < 1050 with p < 97 a prime, we provide a completely normal primitive polynomial of degree n of minimal weight over the field Fp. Any root of such a polynomial will generate a completely normal primitive basis of Fpn over Fp. We have also conjectured a refinement of the primitive normal basis theorem for finite fields and, in addition, we raise several open problems.

Original languageEnglish (US)
Pages (from-to)21-43
Number of pages23
JournalUtilitas Mathematica
StatePublished - 1996

All Science Journal Classification (ASJC) codes

  • Statistics and Probability
  • Statistics, Probability and Uncertainty
  • Applied Mathematics


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