Elliptic curves are fundamental objects in modern algebra and number theory, with applications ranging from cryptography to complex analysis. The structure of the group of points on an elliptic curve can be significantly influenced by the underlying field in which the curve is defined. This study explores the impact of field extensions on the group structure of elliptic curves, focusing on how different types of extensions—such as quadratic, cubic, and higher-order extensions—affect the curve's properties.

We analyze the changes in the torsion subgroups and the overall group structure of elliptic curves as fields are extended. Through both theoretical analysis and computational experiments, we identify patterns and key characteristics that emerge as a result of field extensions. Our results demonstrate that while certain field extensions can simplify the group structure, others introduce additional complexity, influencing the curve’s cryptographic and algebraic applications.

By providing a comprehensive examination of these effects, this study contributes to a deeper understanding of elliptic curve theory and its practical implications in various domains. The findings offer valuable insights for mathematicians and cryptographers working with elliptic curves over extended fields.