Cyclotomic Fields - Relation With Fermat's Last Theorem

Relation With Fermat's Last Theorem

A natural approach to proving Fermat's Last Theorem is to factor the binomial xn + yn, where n is an odd prime, appearing in one side of Fermat's equation

as follows:

xn + yn = (x + y) (x + ζy) … (x + ζn − 1y).

Here x and y are ordinary integers, whereas the factors are algebraic integers in the cyclotomic field Q(ζ_n). If unique factorization of algebraic integers were true, then it could have been used to rule out the existence of nontrivial solutions to Fermat's equation.

Several attempts to tackle Fermat's Last Theorem proceeded along these lines, and both Fermat's proof for n = 4 and Euler's proof for n = 3 can be recast in these terms. Unfortunately, the unique factorization fails in general – for example, for n = 23 – but Kummer found a way around this difficulty. He introduced a replacement for the prime numbers in the cyclotomic field Q(ζ_p), expressed the failure of unique factorization quantitatively via the class number h_p and proved that if h_p is not divisible by p (such numbers p are called regular primes) then Fermat's theorem is true for the exponent n = p. Furthermore, he gave a criterion to determine which primes are regular and using it, established Fermat's theorem for all prime exponents p less than 100, with the exception of the irregular primes 37, 59, and 67. Kummer's work on the congruences for the class numbers of cyclotomic fields was generalized in the twentieth century by Iwasawa in Iwasawa theory and by Kubota and Leopoldt in their theory of p-adic zeta functions.