  The main purpose of this paper is to describe the abelian part $\mathcal
G^{ab}_{K}$ of the absolute Galois group of a global function field $K$ as
pro-finite group. We will show that the characteristic $p$ of $K$ and the non
$p$-part of the class group of $K$ are determined by $\mathcal G^{ab}_{K}$. The
converse is almost true: isomorphism type of $\mathcal G_K^{ab}$ as pro-finite
group is determined by the invariant $d_K$ of the constant field $\mathbb F_q$
introduced in first section and the non $p$-part of the class group.

Number Theory