Untitled

Abstract

Let K be a number field with ring of integers O K . We prove that if 3 does not divide [K : Q] and 3 splits completely in K, then there are no exceptional units in K. In other words, there are no x, y ∈ O × K with x + y = 1. Our elementary 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] has a finite cyclic orbit in O K of length n then n ∈ {1, 2, 4}.