For an odd prime power q, let Fq2=Fq(α), α2=t∈Fq be the quadratic extension of the finite field Fq. In this paper, we consider the irreducible polynomials F(x)=xk-c1xk-1+c2xk-2-⋯-c2qx2+c1qx-1 over Fq2, where k is an odd integer and the coefficients ci are in the form ci=ai+biα with at least one bi≠0. For a given such irreducible polynomial F(x) over Fq2, we provide an algorithm to construct an irreducible polynomial G(x)=xk-A1xk-1+A2xk-2-⋯-Ak-2x2+Ak-1x-Ak over Fq, where the Ai’s are explicitly given in terms of the ci’s. This gives a bijective correspondence between irreducible polynomials over Fq2 and Fq. This fact generalizes many recent results on this subject in the literature.