Density of normal elements
Letf∈ Fq[x] be a monic polynomial of degreen, and let Φ(f) denote the number of polynomials in Fq[x] of degree <nthat are relatively prime tof. Let κ(f) = Φ(f)/qn. We slightly improve the previous known lower bounds of κ(f). The density of normal elements in Fqnover Fqis κ(xn- 1) = Φ(xn- 1)/qn. We prove that κ(xn- 1) ≥C> 0 forn=pe1 1pe2 2···pet t, wherepiare any fixed primes,eivary, andCis a constant independent ofei's. Unfortunately, this is not true for generaln. Indeed, we show an upper bound on κ(xn- 1) for infinitely many values ofnthat goes to 0 asnapproaches infinity. This upper bound is almost tight with our lower bound for a general polynomialf.
|Journal||Finite Fields and their Applications|
Gao, S. (Shuhong), & Panario, D. (1997). Density of normal elements. Finite Fields and their Applications, 3(2), 141–150. doi:10.1006/ffta.1996.0177