Since the advent of class field theory one of the main problems has been its abstract nature that prevented researchers from computing examples. Thus although class field theory classifies abelian extensions of global fields, it failed to provide explicit defining equations for them. Recent progress in computational number theory now permits us to close the gap.

In this talk I will explain how one can find explicit defining equations utilizing Kummer, Artin-Schreier and Witt theory. Applications of these techniques include the explicit construction of good linear codes.

In the end, time permitting, I will also discuss other approaches based on Drinfeld theory.