On the construction of certain odd degree irreducible polynomials over finite fields (Q6651904)

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 the paper under review, the authors consider the irreducible polynomials \N\[\NF(x)=x^k-c_1x^{k-1}+c_2x^{k-2}-\cdots -c_2^qx^2+c_1^qx-1\N\]\Nover \(\mathbb{F}_{q^2}\), 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 \(b_i\neq 0\). For a given such irreducible polynomial \(F(x)\) over \(\mathbb{F}_{q^2}\), the authors provide an algorithm to construct an irreducible polynomial \N\[\NG(x)=x^k-A_1x^{k-1}+A_2x^{k-2}-\cdots -A_{k-2}x^2+A_{k-1}x-A_k\N\]\Nover \(\mathbb{F}_q\), 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.