cyclotomic

Cyclotomic refers to objects in number theory associated with roots of unity, especially cyclotomic polynomials and cyclotomic fields. A cyclotomic polynomial Φ_n(x) is the monic polynomial with integer coefficients whose roots are exactly the primitive nth roots of unity. They are defined by the factorization of x^n − 1 as a product over divisors of n: x^n − 1 = ∏_{d|n} Φ_d(x). Equivalently, Φ_n(x) = ∏_{1 ≤ k ≤ n, gcd(k,n) = 1} (x − e^{2πi k/n}).

Key properties include that deg Φ_n(x) = φ(n), where φ is Euler’s totient function, and Φ_n(x) is irreducible in Z[x] (hence in Q[x]). Coefficients are integers, and while many are small, some have larger magnitudes depending on n. Examples: Φ_1(x) = x − 1; Φ_2(x) = x + 1; Φ_3(x) = x^2 + x + 1; Φ_4(x) = x^2 + 1; Φ_6(x) = x^2 − x + 1. There is a Möbius-function formula, Φ_n(x) = ∏_{d|n} (x^d − 1)^{μ(n/d)}, which encodes a compact arithmetic description.

Cyclotomic also refers to cyclotomic fields, formed by adjoining a primitive nth root of unity ζ_n to the rationals, giving the field Q(ζ_n). The Galois group of Q(ζ_n) over Q is naturally isomorphic to (Z/nZ)×, and the degree of the field extension is φ(n). Cyclotomic concepts are foundational in algebraic number theory, with connections to factorization, class field theory, and modular arithmetic. The term reflects the historic study of division of the circle and roots of unity, a theme dating back to Gauss.