An Intersection of Galois Theory and Elliptic Curves

Author

• Kaylyn Bolaños, Scripps CollegeFollow

Graduation Year

2026

Document Type

Campus Only Senior Thesis

Degree Name

Bachelor of Arts

Department

Mathematics

Reader 1

Christopher Towse

Reader 2

Konrad Aguilar

Rights Information

2026 Kaylyn Bolaños

Abstract

This thesis focuses on providing a base understanding of both Galois theory and elliptic curves, while emphasizing some of their fundamental intersections. Regarding Galois theory, we build up to the idea and definition of a Galois group and work through finding Galois groups for some extension fields. Specifically, we define cyclotomic fields and find the Galois group of (Q(𝜁_5)/Q). We then move on to elliptic curves, with our end goal being to define complex multiplication for our preferred curve 𝐸 : 𝑦^2 = 𝑥^3 + 𝑥. Along the way, we describe the group law for the group of rational points on 𝐸, consider the points on 𝐸 with order dividing 𝑛 denoted by 𝐸[𝑛], and define 𝐸[𝑛] endomorphisms. We use our understanding of both Galois theory and elliptic curves to find the Galois groups of (Q(𝐸[3])/Q) and (Q(𝐸[3])/Q(𝑖)).

Recommended Citation

Bolaños, Kaylyn, "An Intersection of Galois Theory and Elliptic Curves" (2026). Scripps Senior Theses. 2754.
https://scholarship.claremont.edu/scripps_theses/2754