Primitive polynomial with three coefficients prescribed

The authors proved in Fan and Han (Finite Field Appl., in press) that, for any given (a1,a2,a3) Fq3, there exists a primitive polynomial f(x)=xn−σ1xn−1+ +(−1)nσn over Fq of degree n with the first three coefficients σ1,σ2,σ3 prescribed as a1,a2,a3 when n 8. But the methods in Fan and Han (in press) are not effective for the case of n=7. Mills (Existence of primitive polynomials with three coefficients prescribed, J. Algebra Number Theory Appl., in press) resolves the n=7 case for finite fields of characteristic at least 5. In this paper, we deal with the remaining cases and prove that there exists a primitive polynomial of degree 7 over Fq with the first three coefficient prescribed where the characteristic of Fq is 2 or 3.