Irreducible factorizations of polynomials xpk+1−bx+b over a finite field

Abstract

In this paper, we investigate polynomials of the form $f\left(x\right)=x^{p^k+1}-bx+b$, where $0\ne b\in F_{p^n}$, p is a prime, and k divides n. By introducing a new approach based on the projective general linear group, we show that the number of zeros of $f\left(x\right)$ in $F_{p^n}$ belongs to $\{0,1,2,p^k+1\}$, and provide explicit criteria on b for each case. We also count the number of such polynomials corresponding to each possible number of zeros. Moreover, for the cases where $f\left(x\right)$ has at least one zero, we determine its complete irreducible factorization over $F_{p^n}$.