Function field analogue of kummer extensions

Abstract

Let $k$ be a global function field with a fixed prime divisor $\infty$. Let $A$ be the ring of integers outside $\infty$. Using the theory of cyclotomic function fields of rational function fields, F. Schultheis has defined the Calritz-Kummer extensions and computed the factorization of primes in these extensions. In this paper we generalize the work of Schultheis over global function fields. Let $K$ be a finite extension of "cyclotomic" function field $H_{\frak e}^*(\Lambda_\frak m)$. First, We define $Drinfeld$-$Kummer$ extension $K_{\frak m, z}$ over $K$, using a $sgn$-normalized rank 1 Drinfeld $A$-module. And we compute the factorization of primes of $K$ in Drinfeld-Kummer extension $K_{\frak m, z}$.