  We realize Frobenius conjugacy classes in Galois groups of certain
$q$-polynomials over $\mathbb{F}_q(t)$ using specific degree 1 ideals. We
combine this with methods from elementary linear algebra and group theory to
realize transvections in some linear Galois groups. This enables the Galois
group to be identified as a known classical group in several reasonably general
cases.
