Let
K
K
be the function field of a smooth irreducible curve defined over
Q
¯
\overline {Q}
. Let
f
∈
K
[
x
]
f\in K[x]
be of the form
f
(
x
)
=
x
q
+
c
f(x)=x^q+c
, where
q
=
p
r
,
r
≥
1
,
q = p^{r}, r \ge 1,
is a power of the prime number
p
p
, and let
β
∈
K
¯
\beta \in \overline {K}
. For all
n
∈
N
∪
{
∞
}
n\in \mathbb {N}\cup \{\infty \}
, the Galois groups
G
n
(
β
)
=
G
a
l
(
K
(
f
−
n
(
β
)
)
/
K
(
β
)
)
G_n(\beta )=\mathrm {Gal}(K(f^{-n}(\beta ))/K(\beta ))
embed into
[
C
q
]
n
[C_q]^n
, the
n
n
-fold wreath product of the cyclic group
C
q
C_q
. We show that if
f
f
is not isotrivial, then
[
[
C
q
]
∞
:
G
∞
(
β
)
]
>
∞
[[C_q]^\infty :G_\infty (\beta )]>\infty
unless
β
\beta
is postcritical or periodic. We are also able to prove that if
f
1
(
x
)
=
x
q
+
c
1
f_1(x)=x^q+c_1
and
f
2
(
x
)
=
x
q
+
c
2
f_2(x)=x^q+c_2
are two such distinct polynomials, then the fields
⋃
n
=
1
∞
K
(
f
1
−
n
(
β
)
)
\bigcup _{n=1}^\infty K(f_1^{-n}(\beta ))
and
⋃
n
=
1
∞
K
(
f
2
−
n
(
β
)
)
\bigcup _{n=1}^\infty K(f_2^{-n}(\beta ))
are disjoint over a finite extension of
K
K
.