  In this paper we present a combinatorial proof of the Kronecker--Weber
Theorem for global fields of positive characteristic. The main tools are the
use of Witt vectors and their arithmetic developed by H. L. Schmid. The key
result is to obtain, using counting arguments, how many $p$--cyclic extensions
exist of fixed degree and bounded conductor where only one prime ramifies are
there. We then compare this number with the number of subextensions of
cyclotomic function fields of the same type and verify that these two numbers
are the same.
