Irreducible polynomials and linear recurring arrays

For α ϵ Fq the finite field of order q and β ϵ Fq(α.), let Fq(α, β) = Fq(γ). We obtain an explicit formula for the minimal polynomial hgg(x) of γ in terms of the greatest common divisor of two polynomials which are closely related to the minimal polynomials f α(x) of α and gβ(x) of β. We also give an application of this result to linear recurring arrays.