A skew Newton–Puiseux Theorem

We prove a skew generalization of the Newton–Puiseux theorem for the field F=⋃n=1∞C((x1n)) of Puiseux series: For any positive real number α, we consider the ℂ-automorphism σ of F given by x ↦ αx, and prove that every non-constant polynomial in the skew polynomial ring F[t, σ] factors into a product of linear terms. This generalizes the classical theorem where σ = 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.