A note on Galois groups of linearized polynomials

Let $L(X)$ be a monic $q$-linearized polynomial over $F_q$ of degree $q^n$, where $n$ is an odd prime. Recently Gow and McGuire showed that the Galois group of $L(X)/X-t$ over the field of rational functions $F_q(t)$ is $GL_n(q)$ unless $L(X)=X^{q^n}$. The case of even $q$ remained open, but it was conjectured that the result holds too and partial results were given. In this note we settle this conjecture. In fact we use Hensel's Lemma to give a unified proof for all prime powers $q$.