A skew Newton-Puiseux Theorem

Abstract

We prove a skew generalization of the Newton-Puiseux theorem for the field $F = \bigcup_{n=1}^\infty \mathbb{C}((x^\frac{1}{n}))$ of Puiseux series: For any positive real number $\alpha$, we consider the $\mathbb{C}$-automorphism $\sigma$ of $F$ given by $x \mapsto \alpha x$, and prove that every non-constant polynomial in the skew polynomial ring $F[t,\sigma]$ factors into a product of linear terms. This generalizes the classical theorem where $\sigma = {\rm id}$, and gives the first concrete example of a field of characteristic $0$ that is algebraically closed with respect to a non-trivial automorphism -- a notion studied in works of Aryapoor and of Smith. Our result also resolves an open question of Aryapoor concerning such fields. A key ingredient in the proof is a new variant of Hensel's lemma.