Tame Class Field Theory for Global Function Fields

TL;DR

This paper provides a simplified, algebraic proof of class field theory for abelian extensions of global function fields, emphasizing explicit, constructive methods suitable for algorithmic and cryptographic applications.

Contribution

It introduces a new, simplified proof of class field theory for function fields using explicit pairings, bridging number theory and cryptography.

Findings

Simplified algebraic proof of class field theory for function fields

Use of generalized Tate-Lichtenbaum and Ate pairings in proofs

Constructive methods applicable to algorithmic and cryptographic contexts

Abstract

We give a function field specific, algebraic proof of the main results of class field theory for abelian extensions of degree coprime to the characteristic. By adapting some methods known for number fields and combining them in a new way, we obtain a different and much simplified proof, which builds directly on a standard basic knowledge of the theory of function fields. Our methods are explicit and constructive and thus relevant for algorithmic applications. We use generalized forms of the Tate-Lichtenbaum and Ate pairings, which are well-known in cryptography, as an important tool.