On the Density of Normal Bases in Finite Fields

Let Fqn denote the finite field with q^n elements, for q being a prime power. Fqn may be regarded as an n-dimensional vector space over Fq. alpha in Fqn generates a normal basis for this vector space (Fqn : Fq), if {alpha, alpha^q, alpha^q^2 , . . . , alpha^q^(n−1)} are linearly independent over Fq. Let N(q; n) denote the number of elements in Fqn that generate a normal basis for Fqn : Fq, and let nu(q, n) = N(q,n)/q^n denote the frequency of such elements. We show that there exists a constant c > 0 such that nu(q, n) >= c / sqrt(log _q n) ,for all n, q >= 2 and this is optimal up to a constant factor in that we show 0.28477 = 2 We also obtain an explicit lower bound: nu(q, n) >= 1 / e [log_q n], for all n, q >= 2