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

For an odd prime power q, let \(\mathbb {F}_{q^2}=\mathbb {F}_q(\alpha )\) ${\mathbb{F}}_{q^2}={\mathbb{F}}_q(\alpha )$ , \(\alpha ^2=t\in \mathbb {F}_q\) ${\alpha }^2=t\in {\mathbb{F}}_q$ be the quadratic extension of the finite field \(\mathbb {F}_q\) ${\mathbb{F}}_q$ . In this paper, we consider the irreducible polynomials \(F(x)=x^k-c_1x^{k-1}+c_2x^{k-2}-\cdots -c_{2}^qx^2+c_{1}^qx-1\) $F(x)=x^k-c_1{x}^{k-1}+c_2{x}^{k-2}-\cdots -c_2^q{x}^2+c_1^{q}x-1$ over \(\mathbb {F}_{q^2}\) ${\mathbb{F}}_{q^2}$ , where k is an odd integer and the coefficients \(c_i\) $c_i$ are in the form \(c_i=a_i+b_i\alpha \) $c_i=a_i+b_i\alpha$ with at least one \(b_i\ne 0\) $b_i\ne 0$ . For a given such irreducible polynomial F(x) over \(\mathbb {F}_{q^2}\) ${\mathbb{F}}_{q^2}$ , we provide an algorithm to construct an irreducible polynomial \(G(x)=x^k-A_1x^{k-1}+A_2x^{k-2}-\cdots -A_{k-2}x^2+A_{k-1}x-A_k\) $G(x)=x^k-A_1{x}^{k-1}+A_2{x}^{k-2}-\cdots -A_{k-2}{x}^2+A_{k-1}x-A_k$ over \(\mathbb {F}_q\) ${\mathbb{F}}_q$ , where the \(A_i\) $A_i$ ’s are explicitly given in terms of the \(c_i\) $c_i$ ’s. This gives a bijective correspondence between irreducible polynomials over \(\mathbb {F}_{q^2}\) ${\mathbb{F}}_{q^2}$ and \(\mathbb {F}_q\) ${\mathbb{F}}_q$ . This fact generalizes many recent results on this subject in the literature.