Cyclotomic polynomial

The cyclotomic polynomial Φₙ(x) is the monic polynomial whose roots are the primitive nth roots of unity. It has integer coefficients and factors xⁿ − 1 according to root order.

Primitive root of unity: A complex number whose multiplicative order is a specified positive integer. These are exactly the roots of Φₙ(x), distinguishing it from factors for smaller orders.
Polynomial: An expression in variables formed from coefficients and nonnegative integer powers, combined by addition and multiplication. Cyclotomic polynomials are a specialized family defined by their complex roots.
Carl Friedrich Gauss: A German mathematician whose work shaped number theory, algebra, analysis, and geometry. His work on regular polygons established foundational results about cyclotomic equations.
Cyclotomic field: A number field obtained by adjoining a root of unity to the rational numbers. A primitive nth root generates a field whose minimal polynomial is Φₙ(x).
Minimal polynomial: The monic irreducible polynomial over a field that has a specified algebraic element as a root. Φₙ is the minimal polynomial of every primitive nth root over the rational numbers.
Roots of unity: Complex numbers whose positive integer powers equal 1. Cyclotomic polynomials partition these roots according to their exact multiplicative orders.
Divisor: An integer d divides n when n is an integer multiple of d. The factorization of xⁿ − 1 uses one cyclotomic polynomial for every positive divisor d of n.
Disquisitiones Arithmeticae: Gauss's 1801 treatise that organized and advanced the study of number theory. It contains Gauss's analysis of constructible regular polygons through roots of unity.
Regular prime: An odd prime that does not divide the class number of the associated cyclotomic field. Cyclotomic fields and their arithmetic underpin Kummer's criterion for regular primes.
Chebyshev polynomial: A family of polynomials characterized by trigonometric identities and recurrence relations. Chebyshev polynomials encode cosine values, while cyclotomic polynomials encode exact root orders.

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