## Abstract

For q a prime power and n ≥ 2 an integer we consider the existence of completely normal primitive elements in finite field F_{q}n. Such an element α ∈ F_{q}n simultaneously generates a normal basis of F_{q}n over all subfields F_{q}d where d divides n. In addition, α multiplicatively generates the group of all nonzero elements of F_{q}n. For each p^{n} < 10^{50} with p < 97 a prime, we provide a completely normal primitive polynomial of degree n of minimal weight over the field F_{p}. Any root of such a polynomial will generate a completely normal primitive basis of F_{p}n over F_{p}. We have also conjectured a refinement of the primitive normal basis theorem for finite fields and, in addition, we raise several open problems.

Original language | English (US) |
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Pages (from-to) | 21-43 |

Number of pages | 23 |

Journal | Utilitas Mathematica |

Volume | 49 |

State | Published - 1996 |

## All Science Journal Classification (ASJC) codes

- Statistics and Probability
- Statistics, Probability and Uncertainty
- Applied Mathematics