| Description |
Diophantine equations and their solution sets are prominent subjects of study in number theory. These equations are often studied modulo prime numbers or prime ideals in field extensions. Galois Theory is well-suited to study field extensions, but Galois groups are often mysterious. To remedy this, Representation Theory provides a way to study these Galois groups through linear algebra. Elliptic curves are geometric objects that give rise to Galois representations, allowing us to study Galois groups via the properties of elliptic curves and matrices with rational and l-adic number entries. Famous theorems such as Fermat's Last Theorem have been proven using these techniques. This paper provides background in elliptic curves, Galois theory, and representation theory to prove the l-adic Galois representations arising from elliptic curves over Q are irreducible for any prime. This demonstrates one aspect of the relationship between Galois groups, prime numbers, and elliptic curves. |
