  We show that integral monodromy groups of Kloosterman $\ell$-adic sheaves of
rank $n\ge 2$ on $\mathbb{G}_m/\mathbb{F}_q$ are as large as possible when the
characteristic $\ell$ is large enough, depending only on the rank. This variant
of Katz's results over $\mathbb{C}$ was known by works of Gabber, Larsen, Nori
and Hall under restrictions such as $\ell$ large enough depending on
$\operatorname{char}(\mathbb{F}_q)$ with an ineffective constant, which is
unsuitable for applications. We use the theory of finite groups of Lie type to
extend Katz's ideas, in particular the classification of maximal subgroups of
Aschbacher and Kleidman-Liebeck. These results will apply to study reductions
of hyper-Kloosterman sums in forthcoming work.
