Factorization of skew polynomials over k((u))

TL;DR

This paper investigates the structure and factorization of skew polynomials over Laurent series fields in characteristic p, introducing new methods to classify irreducible elements and extend Newton polygon theory.

Contribution

It provides a comprehensive description of irreducible skew polynomials, develops an analogue of Newton polygons, and classifies similarity classes in this setting.

Findings

Characterization of irreducible skew polynomials

Development of Newton polygon analogue

Classification of similarity classes

Abstract

Let $k$ be a perfect field of characteristic $p > 0$, and let $K = k((u))$ be the field of Laurent series over $K$. We study the skew polynomial ring $K[T, \Phi]$, where $\Phi$ is an endomorphism of $K$ that extends a Frobenius endomorphism of $k$. We give a description of the irreducible skew polynomials, develop an analogue of the theory of the Newton polygon in this context, and classify the similarity classes of irreducible elements.