The Effect of Field Extension on the Group Structure of Elliptic Curves

Abstract

An elliptic curve E defined over a finite field K, E(K) is the set of solutions to the general Weierstrass polynomial E: y 2 + a1xy + a3y = x 3 + a2x 2 + a4x + a6 where the coefficients a1, a2, a3, a4, a6 є K. There exist a well defined addition of points on each curve such that the points form an abelian group under the addition operation. This group is either cyclic or isomorphic to the product of two cyclic groups. These set of solutions that form the group lie in the closure of the field K over which the curve is defined. If we allow the set to lie only in a particular extension of K, the addition operation is well defined there too. Therefore we can associate a group to every extension K' of the field K denoted by E(K'). Will the structure of the group defined over the base field K, be affected if the same group is made to lie in the extension K' of K?