Irreducible factorizations of polynomials x^(p^k)+1−bx+b over a finite field----Morningside Center of Mathematics Chinese Academy of Sciences

Irreducible factorizations of polynomials x^(p^k)+1−bx+b over a finite field

Qin Yue

2026-09-14 13:00-14:00

MCM 610

Speaker: Prof. Qin Yue (Nanjing University of Aeronautics and Astronautics)

Title: Irreducible factorizations of polynomials x^(p^k)+1−bx+b over a finite field

Time: 13:00-14:00 September 14, 2026 (Monday)

Inviter: Prof. Ye Tian (MCM)

Place: MCM 610

Abstract: In this paper, we investigate polynomials of the form f(x)=x^(p^k)+1−bx+b, where 0 \neq b∈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(x) 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(x) has at least one zero, we determine its complete irreducible factorization over F_{p^n}.