There are no exceptional units in number fields of degree prime to 3 where 3 splits completely

Abstract

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\} .