Density of Normal Elements

Letf q[x] be a monic polynomial of degreen, and let Φ(f) denote the number of polynomials in q[x] of degree 0 forn=pe11pe22• • •pett, wherepiare any fixed primes,eivary, andCis a constant independent ofeiʹs. 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.