Primitive normal polynomials with the specified last two coefficients

We get that for any element a∈F_q and any primitive element , there exists a primitive normal polynomial f(x)=xⁿ−σ₁xⁿ⁻¹+?+(−1)ⁿ⁻¹σ_n−1x+(−1)ⁿσ_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.