TITLE: Maximal unramified extension of a global function field
QUESTION [5 upvotes]: Can we explicitly describe the unramified extensions of a global function field, for instance $\mathbb{F}_q(T)$?

REPLY [2 votes]: The idelic approach to class field theory is not usually considered "explicit".  A very approachable paper describing completely explicit generation of abelian extensions of a global function field is in 
David R. Hayes, "Explicit class field theory in global function fields"
If I remember correctly, he give you a way to control the conductors, so you should be able to identify the unramified extensions. If you want to read this paper, I suggest starting with Hayes's paper "Explicit class field theory for rational function fields".
Hayes's paper is an alternative to studying the work of Drinfel'd, who obtained the same results in a much more general setting using the machinery of schemes.  This work is analogous to the explicit generation of abelian extensions of imaginary quadratic number fields using the theory of complex multiplication.