Primitive polynomial with three coefficients prescribed

The authors proved in Fan and Han (Finite Field Appl., in press) that, for any given <f>(a1,a2,a3)∈Fq3</f>, there exists a primitive polynomial <f>f(x)=xn-σ1xn-1+⋯+(-1)nσn</f> over <f>Fq</f> of degree <f>n</f> with the first three coefficients <f>σ1,σ2,σ3</f> prescribed as <f>a1,a2,a3</f> when <f>n⩾8</f>. But the methods in Fan and Han (in press) are not effective for the case of <f>n=7</f>. Mills (Existence of primitive polynomials with three coefficients prescribed, J. Algebra Number Theory Appl., in press) resolves the <f>n=7</f> 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 <f>7</f> over <f>Fq</f> with the first three coefficient prescribed where the characteristic of <f>Fq</f> is 2 or 3. [Copyright &y& Elsevier]