On Abelianized Absolute Galois Group of Global Function Fields

Abstract

The main purpose of this paper is to describe the abelian part ${\mathcal{G}}_K^{ab}$ 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 ${\mathcal{G}}_K^{ab}$. 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.

26 pages

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