Hellenica World

Cyclotomic field

The 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.

Properties

A cyclotomic field is the splitting field of the polynomial

xⁿ − 1

and therefore it is a Galois extension of the field of rational numbers. The degree of the extension

[Q(ζ_n):Q]

is given by φ(n) where φ is Euler's phi function. A complete set of Galois conjugates is given by { (ζ_n)^a }, where a runs over the set of invertible residues modulo n (so that a is relative prime to n). The Galois group is naturally isomorphic to the multiplicative group

(Z/nZ)×

of invertible residues modulo n, and it acts on the primitive nth roots of unity by the formula

b: (ζ_n)^a → (ζ_n)^ab.

Relation with regular polygons

Gauss made early inroads in the theory of cyclotomic fields, in connection with the geometrical problem of constructing a regular polygon with a compass and straightedge. His surprising result that had escaped his predecessors was that a regular heptadecagon (with 17 sides) could be so constructed. More generally, if p is a prime number, then a regular p-gon can be constructed if and only if p is a Fermat prime. The geometric problem for a general n can be reduced to the following question in Galois theory: can the nth cyclotomic field be built as a sequence of quadratic extensions?
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

xⁿ + yⁿ = zⁿ

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(ζn). 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 hp and proved that if hp 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.