Density of Normal Elements

Let f 2 Fq [x] be a monic polynomial of degree n, and let (f) denote the number of polynomials in Fq [x] of degree < n that are relatively prime to f . Let (f) = (f)=qn. We slightly improve the previous known lower bounds of (f). The density of normal elements in Fqn over Fq is (xn 1) = (xn 1)=qn. We prove that (xn 1) C > 0 for n = pe1 1 pe2 2 pet t where pi are any xed primes, ei vary, and C is a constant independent of ei's. Unfortunately, this is not true for general n. Indeed, we show an upper bound on (xn 1) for in nitely many values of n that goes to 0 as n approaches in nity. This upper bound is almost tight with our lower bound for a general polynomial f . 3