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.

对于奇素数幂q ，令\(\mathbb {F}_{q^2}=\mathbb {F}_q(\alpha )\) , \(\alpha ^2=t\in \mathbb {F} _q\)是有限域\(\mathbb {F}_q\)的二次扩展。在本文中，我们考虑不可约多项式\(F(x)=x^k-c_1x^{k-1}+c_2x^{k-2}-\cdots -c_{2}^qx^2+c_{ 1}^qx-1\)在\(\mathbb {F}_{q^2}\)上，其中k是奇数整数，系数\(c_i\) 的形式为\(c_i=a_i+b_i \alpha \)至少有一个\(b_i\ne 0\) 。对于给定的这样的不可约多项式F ( x ) 在\(\mathbb {F}_{q^2}\)上，我们提供了一种算法来构造不可约多项式\(G(x)=x^k-A_1x^{ k-1}+A_2x^{k-2}-\cdots -A_{k-2}x^2+A_{k-1}x-A_k\)超过\(\mathbb {F}_q\) ，其中\(A_i\)是根据\(c_i\)明确给出的。这给出了\(\mathbb {F}_{q^2}\)和\(\mathbb {F}_q\)上的不可约多项式之间的双射对应关系。这一事实概括了文献中关于该主题的许多最新结果。