Primitive normal polynomials with the specified last two coefficients

We get that for any element a ∈ F q and any primitive element b ∈ F q ∗ , there exists a primitive normal polynomial f ( x ) = x n − σ 1 x n − 1 + ⋯ + ( − 1 ) n − 1 σ n − 1 x + ( − 1 ) n σ n with σ n − 1 = a , σ n = b for n ≥ 5 . The estimates of hybrid extended Kloostermann sums over finite fields and some new variations of Cohen’s sieve techniques help to get the above result.