Unramified extensions and geometric $\mathbb{Z}_p$-extensions of global function fields

TL;DR

This paper investigates unramified extensions and constructs geometric  d extbackslash Z extsubscript{p}-extensions of global function fields, extending known results and providing new constructions in the area.

Contribution

It extends Perret's results on the ideal class group and introduces a novel construction of geometric  d extbackslash Z extsubscript{p}-extensions with specific properties.

Findings

Extended Perret's result on ideal class groups.

Constructed a geometric  d extbackslash Z extsubscript{p}-extension with particular properties.

Provided new insights into unramified extensions of global function fields.

Abstract

We study on finite unramified extensions of global function fields (function fields of one valuable over a finite field). We show two results. One is an extension of Perret's result about the ideal class group problem. Another is a construction of a geometric $\mathbb{Z}_p$-extension which has a certain property.