Cyclotomic field

In number theory, a cyclotomic field is a number field obtained by adjoining a complex root of unity to Q, the field of rational numbers.

Cyclotomic fields played a crucial role in the development of modern algebra and number theory because of their relation with Fermat's Last Theorem. It was in the process of his deep investigations of the arithmetic of these fields (for prime n) – and more precisely, because of the failure of unique factorization in their rings of integers – that Ernst Kummer first introduced the concept of an ideal number and proved his celebrated congruences.

For n ≥ 1, let ζn = ei/nC; this is a primitive nth root of unity. Then the nth cyclotomic field is the extension Qn) of Q generated by ζn.

Gauss made early inroads in the theory of cyclotomic fields, in connection with the problem of constructing a regular n-gon with a compass and straightedge. His surprising result that had escaped his predecessors was that a regular 17-gon could be so constructed. More generally, for any integer n ≥ 3, the following are equivalent:

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

Here x and y are ordinary integers, whereas the factors are algebraic integers in the cyclotomic field Q(ζn). If unique factorization holds in the cyclotomic integers Z[ζn] , then it can be 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. The complete list of n for which Q(ζn) has unique factorization is[2]

Kummer found a way to deal with the failure of unique factorization. He introduced a replacement for the prime numbers in the cyclotomic integers Z[ζn], measured the failure of unique factorization via the class number hn and proved that if hp is not divisible by a prime p (such 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 established Fermat's theorem for all prime exponents p less than 100, except for 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.