On the construction of certain odd degree irreducible polynomials over finite fields

For an odd prime power q, let \(\mathbb {F}_{q^2}=\mathbb {F}_q(\alpha )\), \(\alpha ^2=t\in \mathbb {F}_q\) be the quadratic extension of the finite field \(\mathbb {F}_q\). In this paper, we consider the irreducible polynomials \(F(x)=x^k-c_1x^{k-1}+c_2x^{k-2}-\cdots -c_{2}^qx^2+c_{1}^qx-1\) over \(\mathbb {F}_{q^2}\), where k is an odd integer and the coefficients \(c_i\) are in the form \(c_i=a_i+b_i\alpha \) with at least one \(b_i\ne 0\). For a given such irreducible polynomial F(x) over \(\mathbb {F}_{q^2}\), we provide an algorithm to construct an irreducible polynomial \(G(x)=x^k-A_1x^{k-1}+A_2x^{k-2}-\cdots -A_{k-2}x^2+A_{k-1}x-A_k\) over \(\mathbb {F}_q\), where the \(A_i\)’s are explicitly given in terms of the \(c_i\)’s. This gives a bijective correspondence between irreducible polynomials over \(\mathbb {F}_{q^2}\) and \(\mathbb {F}_q\). This fact generalizes many recent results on this subject in the literature.