Density of Normal Elements

Authors:
Shuhong Gao;Daniel Panario
Affiliations:
Department of Mathematical Sciences, Clemson University, Clemson, South Carolina, 29634-1907, f1sgao@math.clemson.eduf1;Department of Computer Science, University of Toronto, Toronto, Canada, M5S 1A4f2daniel@cs.toronto.eduf2
Venue:
Finite Fields and Their Applications
Year:
1997

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Abstract

Letf@? F"q[x] be a monic polynomial of degreen, and let @F(f) denote the number of polynomials in F"q[x] of degree =C 0 forn=p^e^"^1"1p^e^"^2"2...p^e^"^t"t, wherep"iare any fixed primes,e"ivary, andCis a constant independent ofe"i's. Unfortunately, this is not true for generaln. Indeed, we show an upper bound on @k(x^n- 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.