Explicit class field theory for rational function fields

Abstract

Developing an idea of Carlitz, I show how one can describe explicitly the maximal abelian extension of the rational function field over F q {{\mathbf {F}}_q} (the finite field of q elements) and the action of the idèle class group via the reciprocity law homomorphism. The theory is closely analogous to the classical theory of cyclotomic extensions of the rational numbers.