Kummer's theory of ideal numbers is one of the foundations of algebraic number theory. In future blogs, I will talk about some of the other very important proofs that came out at this time (impossibility of a general method for quintic equations, transcendence of π, and the fundamental theorem of algebra) and show how Dedekind reinterpreted many of these developments into the modern concepts of ideals, rings, groups, and fields.
Kummer's proof comes down to three major points.
(A) For certain primes (which Kummer called "regular primes"), cyclotomic integers can be said to have a form of unique factorization. [See here for discussion on ideal numbers and how they "save" unique factorization for cyclotomic integers]
(B) For a regular prime λ, there is no solution to xλ + yλ = zλ where x,y,z are pairwise relatively prime all prime to λ
(C) For a regular prime λ, there is no solution to xλ + yλ = zλ where x,y, z are pairwise relatively prime and where λ divides z.
For the full proof, go here.
- Harold M. Edwards, Fermat's Last Theorem