Kummer’s Lemma and Fermat’s Last Theorem

Abstract

The present thesis is devoted to the study and proof of Fermat’s Last Theorem (FLT) for regular primes, with a specific focus on the more intricate Case II. Following the historical and mathematical framework established by Ernst Kummer, the core of this proof relies on a profound intermediate result known as Kummer’s Lemma. While the statement of this lemma is deceptively simple, its rigorous justification demands a considerable amount of preliminary mathematical machinery.

There is not a single way to prove Kummer’s Lemma. One well-known approach relies on the advanced architecture of class field theory; however, developing that theory from the ground up would require an extensive detour, occupying many pages of dense discussion. Consequently, we have chosen an alternative, more analytical path. This route leans heavily on the elegant framework of p-adic analysis, specifically p-adic L-functions and p-adic regulators, drawing significant inspiration from Lawrence C. Washington’s foundational text, Introduction to Cyclotomic Fields.

Because Washington’s book is a comprehensive treatise on the broader theory of cyclotomic fields rather than a specialized text on Fermat’s Last Theorem, extracting the exact sequence of results needed for Kummer’s Lemma can be a daunting task, and there is a lack of pedagogical literature dedicated specifically to unpacking this proof in detail. The primary objective of this project, therefore, is to streamline and clarify this theory. By isolating and presenting strictly the necessary analytical and algebraic tools, this thesis aims to provide a self-contained and direct pathway to the proof of Kummer’s Lemma and, ultimately, the second case of Fermat’s Last Theorem for regular primes.