Ireducible quadratic factors of \(x^{(q^n+1)/2}+ax+b\) over \(\mathbb{F}_q\) (Q1383498)

The factorization of special families of polynomials have been studied in some papers, e.g. in the classical work of \textit{O. Ore} [Trans. Am. Math. Soc. 36, 243-274 (1934; Zbl 0009.10003)]. In the present paper, the authors consider polynomials of the form \(f_{a,b,n}=x^{(q^n+1)/2} + ax +b\) over the finite field \(F_q\) with \(q\) elements and study whether \(f_{a,b,n}\) admits irreducible quadratic factors. The main result is as follows: Assume that \(b\) is nonzero and that \(q\) is odd. If \(n\) is even or \(a^2+1\) is a square in \(F_q\), then \(f_{a,b,n}\) does not have an irreducible quadratic factor over \(F_q\). On the other hand, if \(f_{a,b,n}\) does have a monic irreducible quadratic factor over \(F_q\), then it is unique and equal to \(g:=x^2 + 2(b/a)x + b^2 / (a^2+1)\). A condition is also derived in terms of quadratic and biquadratic symbols for \(g\) dividing \(f_{a,b,n}\).