TY - JOUR
AU - Gudmund Frandsen
PY - 1997/06/14
Y2 - 2020/11/30
TI - On the Density of Normal Bases in Finite Fields
JF - BRICS Report Series
JA - BRICS
VL - 4
IS - 44
SE - Articles
DO - 10.7146/brics.v4i44.18970
UR - https://tidsskrift.dk/brics/article/view/18970
AB - 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 forFqn : 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 thatnu(q, n) >= c / sqrt(log _q n) ,for all n, q >= 2and this is optimal up to a constant factor in that we show0.28477 = 1 / e [log_q n], for all n, q >= 2
ER -