Mixed Characteristic Artin Schreier Polynomials

TL;DR

This paper generalizes Artin-Schreier polynomials to mixed characteristic settings, providing a method to lift cyclic extensions over local rings with arbitrary characteristic, and introduces new algebraic structures related to Galois actions.

Contribution

It introduces a generalized polynomial framework for cyclic extensions in mixed characteristic and constructs a generic Galois extension to facilitate lifting results.

Findings

Generalized Artin-Schreier polynomial for mixed characteristic

Construction of a generic Galois extension for lifting

Description of cyclic algebras and new algebraic structures

Abstract

We present here a version of the Artin-Schreier polynomial that works in any characteristic. Let $C_p$ be the cyclic group of prime order $p$. Equivalently, we prove one can lift degree $p$ cyclic $C_p$ extensions over local rings $R,M$ where $R/M$ has characteristic $p$ and $R$ has arbitrary characterstic. Let $\rho \in \Bbb C$ be a primitive $p$ root of one. We first consider the case $R$ has a primitive $p$ root of one, by which we mean that there is a given homomorphism $f: \Bbb Z[\rho] \to R$. In this context we can write a specific polynomial which generalizes the Artin-Schreier polynomial. We next consider arbtitrary $R$ and construct a mixed characteristic "generic" Galois extension that proves the lifting result, but here we do not supply a polynomial. It is useful to view these results in terms of Galois actions. If $C_p$ is generated by $\sigma$, then Artin-Schreier polynomials describe $C_p$ Galois extensions generated by elements $\theta$ where $\sigma(\theta) = \theta + 1$. If $\rho' = f(\rho)$, then our lifted Galois action is $\sigma(\theta) = \rho'\theta + 1$. In section two we descend this extension to rings without root of one. In section three we use this new description of cyclic extensions, give the obvious description of the corresponding cyclic algebras, and then define a new algebra which specializes to general differential crossed products in characteristic $p$ but which are cyclic in characteristic not $p$. The algebra in question is generated by $x,y$ subject to the relation $xy - \rho{yx} = 1$.