Abstract
Let Fqn denote the finite field with qn elements, for q a prime power. Fqn may be regarded as an n-dimensional vector space over Fq. α∈Fqn generates a normal basis for this vector space (Fqn:Fq), if {α, αq, αq2 , … , αqn−1} are linearly independent over Fq. Let Nq(n) denote the number of elements in Fqn that generate a normal basis for Fqn:Fq, and let νq(n)=Nq(n)/qn denote the frequency of such elements. We show that there exists a constant c>0 such that and this is optimal up to a constant factor in that we showWe also obtain an explicit lower bound:
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