Factorization of skew polynomials over k((u))

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.