Sunday, May 07, 2006

Fermat's Last Theorem: Proof for regular primes

One of the highpoints of the 19th century mathematics is Kummer's proof of Fermat's Last Theorem for regular primes.

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.



Lieven said...

Unless the detailed proof handles this, don't you need a case (III) where lambda divides x?

Larry Freeman said...

Hi Lieven,

No, you don't need to worry about the case you mention.

There are two reasons:

(1) You can always assume that x,y,z are relatively prime. (see here for details)

(2) x^n + y^n = z^n is symmetrical, so let's assume that lambda divides x and n is odd (see here for why we can assume that n is odd)

Let's define x' = (-x) and z' = (-z), then we have:

(x')^n = y^n + (z')^n

So that we have the same form where lamdba divides (x')^n.