Irreducible factors of psi-polynomials over finite fields (Q1111610)

If \({\mathbb F}_ q\) denotes the finite field of order \(q\), let \(C=\left( \begin{matrix} \alpha \beta \\ \gamma \delta \end{matrix} \right)\) be an invertible \(2\times 2\) matrix over \({\mathbb F}_ q\). By a psi-polynomial over \({\mathbb F}_ q\) is meant a polynomial of the form \(\psi_{C,k}(x)=\gamma x^{q^ k+1}+\delta x^{q^ k}-\alpha x- \beta\) for \(k=0,1,\ldots\). Psi-polynomials have been extensively studied in the literature. The C-invariant irreducible polynomials over \({\mathbb F}_ q\) are described as the irreducible factors of the psi-polynomials. The author studies the factorization of the psi-polynomials into components, where a component is defined as the product of all monic, irreducible factors of a fixed degree. The author obtains numerous results concerning the components of the psi-polynomials. In particular, the degrees of the irreducible factors of \(\psi_{C,k}\) are determined along with their number. The results are however too complicated to be stated here.