The Carlitz Algebras

Abstract

The Carlitz ${\mathbb{F}}_q$-algebra $C=C_{\nu }$, $\nu \in \mathbb{N}$, is generated by an algebraically closed field $\CK $ (which contains a non-discrete locally compact field of positive characteristic $p>0$, i.e. $K\simeq {\mathbb{F}}_q[[x,x^{-1}]]$, $q=p^{\nu }$), by the (power of the) {\em Frobenius} map $X=X_{\nu }:f\mapsto f^q$, and by the {\em Carlitz derivative} $Y=Y_{\nu }$. It is proved that the Krull and global dimensions of $C$ are 2, a classification of simple $C$-modules and ideals are given, there are only {\em countably many} ideals, they commute $(IJ=JI)$, and each ideal is a unique product of maximal ones. It is a remarkable fact that any simple $C$-module is a sum of eigenspaces of the element $YX$ (the set of eigenvalues for $YX$ is given explicitly for each simple $C$-module). This fact is crucial in finding the group $\Aut_{\Fq}(C)$ of $\Fq$-algebra automorphisms of $C$ and in proving that two distinct Carlitz rings are not isomorphic $(C_{\nu }\not\simeq C_{\mu }$ if $\nu \ne \mu$). The centre of $C$ is found explicitly, it is a UFD that contains {\em countably many} elements.

16 pages