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| Let q be a prime or prime power and F_{q^{n}} the extension of q elements finite field F_{q} with degree n(n>1). Davenport, Lenstra and Schoof proved that there exists a primitive element \\alpha\\in F_{q^{n}}such that \\alpha generates a normal basis of F_{q^{n}} over
F_{q}. Later, Mullin, Gao and Lenstra, etc., raised the definition of optimal normal bases and constructed such bases. In this paper, we determine all primitive type I optimal normal bases and all finite fields in which there
exists a pair of reciprocal elements \\alpha and \\alpha^{-1} such that both of them generate optimal normal bases of F_{q^{n}} over F_{q}. Furthermore, we obtain a sufficient condition for the existence of primitive type II optimal normal bases over finite fields and prove that all primitive optimal normal elements are conjugate to each other. |
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| Keywords:Finite fields;Normal bases;Primitive elements;Optimal normal bases |
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