[1] 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.
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.
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 as follows: 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.
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.