Torsion points of low order on elliptic curves and Drinfeld modules

In this thesis. we calculate the Galois groups of extensions generated by torsion points of low order on elliptic curves and Drinfeld modules through their corresponding division polynomials. We investigate division po1:ynomials of degree up to and including four, which correspond to 2-torsion and 3-torsj.on points on elliptic curves and (T+a)- torsion and (T2 + aT + b)-torsion points on Drinfeld modules of rank 1 .and 2. These calculations depend on the invariants that classify elliptic curves and Drinfeld modules up to an isomorphism.