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 $n \in \{1, 2, 4\}$.