Irreducible polynomials and linear recurring arrays

For alpha is an element of F-q the finite field of order q and beta is an element of F-q(alpha), let F-q(alpha,beta) = F-q(gamma). We obtain an explicit formula for the minimal polynomial h(gamma)(x) of gamma in terms of the greatest common divisor of two polynomials which are closely related to the minimal polynomials f(alpha)(x) of alpha and g(beta)(x) of p. we also give an application of this result to linear recurring arrays.