Abstract
Let k be the rational function field, and A be the polynomial ring. Let ψ be a Drinfeld module over a finite extension L of k which is viewed as an A-field of generic characteristic. Via ψ, L becomes an A-module. We denote this module by ψ(L). Denote Λ(M,ψ) to be the module of M-torsion points of ψ. For finitely generated A-submodule Γof ψ(L), set H(Γ,M) to be the Galois group Gal(L(Λ(M,ψ),1/MΓ)/L(Λ(M,ψ) )).In this thesis, we establish the Kummer theory of division points over the Carlitz module, Drinfeld modules of rank one, singular Drinfeld modules of rank 2, and more generally, singular Drinfeld modules of any rank. Under some mild conditions, it can be shown that for almost all primes p in A, the Galois group H(Γ,p) is as large as possible. In particular, if Γ is free of rank r, then H(Γ,p) is isomorphic to the direct product of the r copies of Λ(p,ψ).