Infinite class towers for function fields

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 $\mathbb{F}_q[t]$.