🤖 AI Summary
This work presents the first complete formalization in Lean 4 of the regular prime case of Fermat’s Last Theorem. The central challenge—formalizing Kummer’s Lemma—was addressed not via modern class field theory, but through a novel constructive approach grounded in Hilbert’s Theorems 90–94, yielding a fully machine-verifiable proof. The formalization establishes a rigorous, end-to-end chain from foundational ring theory to the key lemma, with every intermediate step explicitly verified by the theorem prover. The underlying algebraic number theory library, together with the adopted proof strategy, substantially enhances both the feasibility and trustworthiness of high-level number-theoretic formalization. This effort sets a new benchmark for deep, rigorous formalization of landmark theorems in number theory.
📝 Abstract
We formalize a complete proof of the regular case of Fermat's Last Theorem in the Lean4 theorem prover. Our formalization includes a proof of Kummer's lemma, that is the main obstruction to Fermat's Last Theorem for regular primes. Rather than following the modern proof of Kummer's lemma via class field theory, we prove it by using Hilbert's Theorems 90-94 in a way that is more amenable to formalization.