Cyclotomic field

In number theory, a cyclotomic field is a number field obtained by adjoining a complex primitive root of unity to Q, the field of rational numbers. The n-th cyclotomic field Q(ζ_n) (where n > 2) is obtained by adjoining a primitive n-th root of unity ζ_n to the rational numbers.

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.

A cyclotomic field is the splitting field of the cyclotomic polynomial

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

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

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