  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 $q$.
