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 = e2πi/n ∈ C; this is a primitive nth root of unity. Then the nth cyclotomic field is the extension Q(ζn) of Q generated by ζn. The nth cyclotomic polynomial is irreducible, so it is the minimal polynomial of ζn over Q. The conjugates of ζn in C are therefore the other primitive nth roots of unity: ζ for 1 ≤ k ≤ n with gcd(k, n) = 1. The degree of Q(ζn) is therefore [Q(ζn) : Q] = deg Φn = φ(n), where φ is Euler's totient function. The roots of xn − 1 are the powers of ζn, so Q(ζn) is the splitting field of xn − 1 (or of Φ(x)) over Q. Therefore Q(ζn) is a Galois extension of Q. The Galois group is naturally isomorphic to the multiplicative group , which consists of the invertible residues modulo n, which are the residues a mod n with 1 ≤ a ≤ n and gcd(a, n) = 1. The isomorphism sends each to a mod n, where a is an integer such that σ(ζn) = ζ. The ring of integers of Q(ζn) is Z[ζn]. For n > 2, the discriminant of the extension Q(ζn) / Q is In particular, Q(ζn) / Q is unramified above every prime not dividing n. If n is a power of a prime p, then Q(ζn) / Q is totally ramified above p. If q is a prime not dividing n, then the Frobenius element corresponds to the residue of q in . The group of roots of unity in Q(ζn) has order n or 2n, according to whether n is even or odd. The unit group Z[ζn]× is a finitely generated abelian group of rank φ(n)/2 – 1, for any n > 2, by the Dirichlet unit theorem.