Primitive normal polynomials with the specified last two coefficients

We get that for any element a∈Fqa∈Fq and any primitive element b∈Fq∗, there exists a primitive normal polynomial f(x)=xn−σ1xn−1+⋯+(−1)n−1σn−1x+(−1)nσnf(x)=xn−σ1xn−1+⋯+(−1)n−1σn−1x+(−1)nσn with σn−1=aσn−1=a, σn=bσn=b for n≥5n≥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.