A skew Newton-Puiseux theorem and algebraically closed fields with respect to an automorphism

A field K is algebraically closed with respect to an automorphism sigma if every non-constant polynomial in the skew polynomial ring K[x, sigma] factors into a product of linear terms. A classical example (Ore, 1933) is the algebraic closure of a finite field, with respect to the Frobenius automorphism. In this talk we give the first explicit example in characteristic 0, via a generalization of the Newton-Puiseux theorem. Namely, we show that the field of Puiseux series is algebraically closed with respect to non-trivial automorphisms. As a consequence, we solve a question of Aryapoor concerning such fields. Joint work with Thieu N. Vo.