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