Mixed Characteristic Artin Schreier Polynomials

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 $xy - \rho{yx} = 1$.