Let
K
K
be a number field with ring of integers
O
K
\mathcal O_{K}
. We prove that if
3
3
does not divide
[
K
:
Q
]
[K:\mathbb Q]
and
3
3
splits completely in
K
K
, then there are no exceptional units in
K
K
. In other words, there are no
x
,
y
∈
O
K
×
x, y \in \mathcal O_{K}^{\times }
with
x
+
y
=
1
x + y = 1
. Our elementary
p
p
-adic proof is inspired by the Skolem-Chabauty-Coleman method applied to the restriction of scalars of the projective line minus three points. Applying this result to a problem in arithmetic dynamics, we show that if
f
∈
O
K
[
x
]
f \in \mathcal O_{K}[x]
has a finite cyclic orbit in
O
K
\mathcal O_{K}
of length
n
n
then
n
∈
{
1
,
2
,
4
}
n \in \{1, 2, 4\}
.