A skew Newton–Puiseux Theorem

Abstract We prove a skew generalization of the Newton–Puiseux theorem for the field $$F=\bigcup\nolimits_{n=1}^{\infty} \, {\mathbb C}((x^{1 \over n}))$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>F</mml:mi> <mml:mo>=</mml:mo> <mml:msubsup> <mml:mo>⋃</mml:mo> <mml:mrow> <mml:mi>n</mml:mi> <mml:mo>=</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> <mml:mrow> <mml:mi>∞</mml:mi> </mml:mrow> </mml:msubsup> <mml:mspace/> <mml:mrow> <mml:mrow> <mml:mi>C</mml:mi> </mml:mrow> </mml:mrow> <mml:mo>(</mml:mo> <mml:mo>(</mml:mo> <mml:msup> <mml:mi>x</mml:mi> <mml:mrow> <mml:mfrac> <mml:mn>1</mml:mn> <mml:mi>n</mml:mi> </mml:mfrac> </mml:mrow> </mml:msup> <mml:mo>)</mml:mo> <mml:mo>)</mml:mo> </mml:math> 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.