On Galois groups of linearized polynomials related to the general linear group of prime degree

Let L(x) be any q-linearized polynomial with coefficients in Fq, of degree qn. We consider the Galois group of L(x) +tx over Fq(t), where t is transcendental over Fq. We prove that when n is a prime, the Galois group is always GL(n, q), except when L(x) = xqn. Equivalently, we prove that the arithmetic monodromy group of L(x)/x is GL(n, q), except when L(x) = xqn, and also equivalently, we prove that the image of the mod-(t) Galois representation of the Drinfeld module arising from L(x) is all of GL(n, q).& COPY; 2023 The Author(s). Published by Elsevier Inc. This is an open access article under the CC BY license (http:// creativecommons .org /licenses /by /4 .0/).