Class groups of characteristic-p function field analogues of Q(n^(1/p))

Abstract

In the theory of cyclotomic function fields, the Carlitz module $\Lambda_M$ associated to a polynomial $M$ in a global function field of characteristic $p$ provides a strong analogy to the roots of unity $\mu_p$ in a number field. In this work, we consider a natural extension of this theory to give a compatible analogue of the $p$-th root of an integer $n$. The most fundamental case, and the one which most closely mimics the number field situation, is when the Carlitz module is defined by a linear polynomial (which can be assumed to be $T$) in $k={\mathbb F}_q(T)$. The Carlitz module $\Lambda_T$ generates a degree-$(q-1)$ extension $k(\Lambda_T)$ which shares many properties with the field ${\mathbb Q}(\mu_p)$, where $\mu_p$ is the module of $p$-th roots of unity. To form the analogue of ${\mathbb Q}(\sqrt[p]{n})$, we define a degree-$q$ extension $F/k$ associated to a polynomial $P(T) \in k$, for which the normal closure is formed by adjoining $\Lambda_T$. In the introduction, we describe in detail the parallels between this construction and that in the number field setting. We then compute the class number $h_F$ for a large number of such fields. The remainder of the work is concerned with proving results about the class groups and class numbers of this family of fields. These are:\begin{itemize} \item a formula relating the class number of $F$ to that of its normal closure, along with a theorem about the structure of the class group of the normal closure \item a formula relating the class number of a compositum of such $F$ to the class numbers of the constituent fields \item conditions on $P(T)$ for when the characteristic, $p$, of $F$ divides its class number, along with bounds on the rank of the $p$-part of the class group. \end{itemize}