Galois Representation on Elliptic Curve

This thesis explores the orders of Galois representations about torsion subgroups of elliptic curves. We review the literature on elliptic curves and Serre's theorem. We describe a field formed by adjoining torsion subgroup of an elliptic curve. We show that the extension is finite and algebraic. Next, we construct a Galois group from the extension and use the relationship with a generalized linear group to find the possible values of the order of the Galois group. The order depends on the field where an elliptic curve is defined, the reducibility of f(x), and structure of the torsion subgroup. This approach provides the same insight as Serre's theorem that provides an upper bound of the order of Galois representation of an extended field given by adjoining a subgroup of points of an elliptic curve.