We study the distribution of the Galois group of a random
q
q
-additive polynomial over a rational function field: For
q
q
a power of a prime
p
p
, let
f
=
X
q
n
+
a
n
−
1
X
q
n
−
1
+
…
+
a
1
X
q
+
a
0
X
f=X^{q^n}+a_{n-1}X^{q^{n-1}}+\ldots +a_1X^q+a_0X
be a random polynomial chosen uniformly from the set of
q
q
-additive polynomials of degree
n
n
and height
d
d
, that is, the coefficients are independent uniform polynomials of degree
deg
a
i
≤
d
\deg a_i\leq d
. The Galois group
G
f
G_f
is a random subgroup of
GL
n
(
q
)
\operatorname {GL}_n(q)
. Our main result shows that
G
f
G_f
is almost surely large as
d
,
q
d,q
are fixed and
n
→
∞
n\to \infty
. For example, we give necessary and sufficient conditions so that
SL
n
(
q
)
≤
G
f
\operatorname {SL}_n(q)\leq G_f
asymptotically almost surely. Our proof uses the classification of maximal subgroups of
GL
n
(
q
)
\operatorname {GL}_n(q)
. We also consider the limits:
q
,
n
q,n
fixed,
d
→
∞
d\to \infty
and
d
,
n
d,n
fixed,
q
→
∞
q\to \infty
, which are more elementary.