Exceptional scatteredness in prime degree

Abstract Let q be an odd prime power and n an integer. Let l ∈ F q n [ x ] be a q-linearised t-scattered polynomial of linearized degree r. Let d = max ⁡ { t , r } be an odd prime number. In this paper we show that under these assumptions it follows that l = x . Our technique involves a Galois theoretical characterization of t-scattered polynomials combined with the classification of transitive subgroups of the general linear group over the finite field F q .