On the Density of Normal Bases in Finite Fields,

Let qn denote the finite field with qn elements, for q a prime power. qn may be regarded as an n-dimensional vector space over q. α qn generates a normal basis for this vector space ( qn: q), if {α, αq, αq2 , … , αqn−1} are linearly independent over q. Let Nq(n) denote the number of elements in qn that generate a normal basis for qn: q, 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 show We also obtain an explicit lower bound: