Infinite class towers for function fields

TL;DR

This paper constructs examples of function fields over finite fields that have infinite Hilbert p-class field towers, extending cyclotomic function fields with specific ramification properties.

Contribution

It provides explicit examples of function fields with infinite class field towers, expanding the understanding of ramification and tower structures in function field arithmetic.

Findings

Existence of function fields with infinite Hilbert p-class towers

Construction of such fields as extensions of cyclotomic function fields

Ramification only at one finite regular prime

Abstract

This paper gives examples of function fields $K_0$ over a finite field $\mathbb{F}_q$ of $p$ power order ramified only at one finite regular prime over $\mathbb{F}_q(t)$, which admit infinite Hilbert $p$-class field towers. Such a $K_0$ can be taken as an extension of a cyclotomic function field $\mathbb{F}_q(t)(\lambda_{\mathfrak{p}^m})$ for a certain regular prime $\mathfrak{p}$ in $\mathbb{F}_q[t]$.