At a meeting of the Paris Academy in 1847 Lame' announced that he had succeeded in proving Fermat's Last Theorem, and was generous enough to admit that he'd gotten the inspiration for the proof from Liouville, who was also in attendance. However, Liouville immediately took the floor to point out that (1) the "inspiration", which involved working with complex "integers" over various fields, was not new, and (2) that Lame' was not justified in assuming unique factorization for these integers. It's true that the ring of Gaussian integers, i.e., numbers of the form a+ib for integers a and b possesses unique factorization, but Lame' was proposing to work with a more general class of numbers. The basic "units" of the Gaussian integers are the 4th roots of 1, which are the numbers 1,i,-1,-i. The numbers Lame' was talking about for a given prime p are called the "p-cyclotomic integers", and have as their basic "units" the pth roots of 1. In the weeks that followed the meeting it transpired that Kummer had already (three years before) published a paper showing the failure of unique factorization in certain cyclotomic fields. The first example where unique factorization fails is with p=23. Kummer soon published his theory of "ideal numbers" which enabled him to recover unique factorization (up to units) for cyclotomic integers, and this, in turn, led to a genuine proof of Fermat's Last Theorem for a large set of prime exponents, although not all. Specifically, Kummer's ideal numbers (the forerunners of Dedekind's "ideals") and his work on the higher reciprocity laws enabled him to prove Fermat's Last Theorem for all exponents p such that the class number h(p) of the cyclotomic integers is not divisible by p itself. If the ring of p-cyclotomic integers has ordinary unique factorization, then h(p)=1, so obviously Kummer's proof applies to all those primes. In addition, it applies to other cases, including p=23, where h(23) exceeds 1 but isn't divisible by 23. A prime such that p does not divide the class number h(p) is called "regular", and Kummer showed that a prime is regular if and only if p does not divide the numerators of the Bernoulli numbers B_n for all even n up to p-3. With this criterion he was able to show that the only "irregular" primes less than 163 (the largest prime p for which the class number of the quadratic field sqrt(-p) equals 1) are 37, 59, 67, 101, 103, 131, 149, and 157. It is known that there exist infinitely many of these irregular primes, but (ironically) we have no proof that there are infinitely many regular primes. At least that was the case a few years ago, and I think it's still true, even post-Wiles. By the way, Singh's popular book on Fermat's Last Theorem seems to confuse many readers on the question of factorization, because he refers to unique factorization being applicable to "real numbers". Presumably he meant to say "rational integers" or words to that effect. Obviously the set of real numbers doesn't possess unique factorization in any useful sense. A pedant might contend that the reals possess unique factorization *up to multiplication by units*, but this is inane, because EVERY real number other than zero is a "unit", i.e., a divisor of 1. It would be more sensible to say that the concept of unique factorization only applies in a meaningful way to sets of numbers that are not all units.

Return to MathPages Main Menu