Character sums over Galois rings and primitive polynomials over finite fields

摘要

Using the estimates of character sums over Galois rings and Cohen's sieve, we prove that there exists a primitive polynomial f(x) = x~n - a_1x~(n-1) + ... + (-1)~n a_n over F_q of degree n with the first three coefficients a_1, a_2, a_3, prescribed when n ≥ 8.