@article{Frandsen_1997, title={On the Density of Normal Bases in Finite Fields}, volume={4}, url={https://tidsskrift.dk/brics/article/view/18970}, DOI={10.7146/brics.v4i44.18970}, abstractNote={<p>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<br />{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<br />Fqn : Fq, and let nu(q, n) = N(q,n)/q^n denote the frequency of such elements.<br />We show that there exists a constant c > 0 such that<br />nu(q, n) >= c / sqrt(log _q n) ,for all n, q >= 2<br />and this is optimal up to a constant factor in that we show<br />0.28477 <= lim inf (q, n) sqrt( log_q n ) <= 0.62521, for all q >= 2</p><p>We also obtain an explicit lower bound:<br /> nu(q, n) >= 1 / e [log_q n], for all n, q >= 2</p>}, number={44}, journal={BRICS Report Series}, author={Frandsen, Gudmund Skovbjerg}, year={1997}, month={Jun.} }