Abstract:An analogue to cyclotomic number fields for function fields over the finite field F_|| with q elements 

was investigated by L. Carlitz in 1935, and has been studied recently  by D. Hayes, M. Rosen, S. Galovich, 

M. Bilhan and others. For each non-zero polynomial M  in F_||[T],we denote by k(L_M) the cyclotomic 
function field associated with M. Replacing T by  in k and considering the cyclotomic function field F_n 
corresponding to , we get an extension of k denoted by L_n which is the fixed field of F_n modulo F_||^*. 
We call a (n,n,M)-extension the composite N = k_n k(L_M) L_n where k_n is the constant field extension of 

degree n over k . By the analogue of Kronecker-Weber theorem for function fields, every finite abelian 

extension  F of  k  is contained in a (n,n,M)-extension N .Then the class number of  F divides the class 

number of  N . In this thesis we give an analytic class number formula for (n,n,M)-extensions 

when M has non-zero constant term. Then we compute the class number for specific (n,n,M)-extensions. 

Key words: Function Fields, Cyclotomic Function Fields, Zeta  Functions, L-Functions, 

Class Number, Abelian Extensions.