Available at: https://digitalcommons.calpoly.edu/theses/3088
Date of Award
6-2025
Degree Name
MS in Mathematics
Department/Program
Mathematics
College
College of Science and Mathematics
Advisor
Eric Brussel
Advisor Department
Mathematics
Advisor College
College of Science and Mathematics
Abstract
This thesis explores the evolution of reciprocity laws in number theory in order to provide a conceptual bridge between the classical ideas of quadratic reciprocity and the modern framework of the Langlands program. We develop the necessary algebraic background to understand how the splitting behavior of primes in number fields reflects deep arithmetic structure in ℚ. Starting with quadratic fields and cyclotomic extensions, we motivate the development of the Kronecker–Weber theorem and the characterization of abelian extensions of ℚ. We then introduce Artin reciprocity and show how it generalizes quadratic reciprocity through the formalism of Frobenius elements and Artin L-functions. These ideas naturally lead into the Langlands philosophy, where the equality of L-functions that arise from representations of two distinct objects (one a Galois representation and the other an automorphic representation) becomes the natural framework for a generalized reciprocity law. We conclude by examining how the modularity theorem for elliptic curves implies Fermat’s Last Theorem and briefly comment on recent developments in the geometric Langlands program.