The unit equation has no solutions in number fields of degree prime to $3$ where $3$ splits completely

Abstract

Let $K$ be a number field with ring of integers ${\mathcal{O}}_K$. We prove that if $3$ does not divide $[K:\mathbb{Q}]$ and $3$ splits completely in $K$, then the unit equation has no solutions in $K$. In other words, there are no $x,y\in {\mathcal{O}}_K^{\times }$ 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 $f\in {\mathcal{O}}_K[x]$ has a finite cyclic orbit in ${\mathcal{O}}_K$ of length $n$ then $n\in \{1,2,4\}$.

4 pages, comments appreciated. Update corrects typos and adds an application