Class Group Relations in a Function Field Analogue of ${\mathbb Q}(\zeta_p, \sqrt[p]{n})$

TL;DR

This paper investigates the class group structure of a function field analogue of a number field involving roots of polynomials, establishing key properties of its divisor class group and class number in relation to the base field.

Contribution

It proves that the degree-0 divisor class group of the Galois closure is a power of a certain group and that the class number is a specific power of the base field's class number, extending classical number field results.

Findings

Degree-0 divisor class group is a^{p-1} for some group a

Class number is (class number of F)^{p-1}

Results mirror classical number field theorems

Abstract

For an odd prime $p$ and polynomial $P(T)$, we consider the extension $F$ of $k={\mathbb F}_p(T)$ defined by adjoining a root of $x^p+Tx-P(T)$. Such a field is a function field analogue of the number field ${\mathbb Q}(\sqrt[p]{n})$. We prove two theorems about the Galois closure $L$ of $F$: that its degree-0 divisor class group is $A^{p-1}$ for some group $A$, and that its class number is the $(p-1)$-st power of the class number of $F$, in analogy with results of R. Schoof and T. Honda for number fields.