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.