Kummer's Objection

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.