Cyclotomic polynomial¶

The monic irreducible integer polynomial whose roots are exactly the primitive nth roots of unity.

Version: v1 · 2026-09-08 · History
Domain-specific #: 4016
Origin domain: algebra
Subdomain: algebra

Core Idea¶

The index n is positive, primitiveness means exact order n and factorization and sign conventions must distinguish Phi_n from x^n minus one. Primitive roots are selected from all nth roots of unity and multiplied into a monic polynomial; Möbius inversion or recursive factorization of x^n−1 yields integer coefficients. The abstraction is therefore identified by a declared carrier, a transformation or constraint over that carrier, and an invariant that tells an analyst whether the named structure is genuinely present.

Scope of Application¶

Cyclotomic polynomial belongs to algebra and is useful where the analyst can specify the typed algebra carrier, including objects, relations, parameters, conventions, evidence, boundaries, and comparison targets, then evaluate the positive integer n, primitive nth roots and coprimality index, product definition, monicity and integer coefficients, irreducibility over rationals, degree Euler phi(n), factorization of x^n minus one, Möbius product and representative examples are explicit. The scope is broad within that domain but bounded by the need for the positive integer n, primitive nth roots and coprimality index, product definition, monicity and integer coefficients, irreducibility over rationals, degree Euler phi(n), factorization of x^n minus one, Möbius product and representative examples are explicit.

Clarity¶

The abstraction clarifies a crowded vocabulary by making the positive integer n, primitive nth roots and coprimality index, product definition, monicity and integer coefficients, irreducibility over rationals, degree Euler phi(n), factorization of x^n minus one, Möbius product and representative examples are explicit the center of the account. A claim should name the carrier, the governing operation or relation, the applicable assumptions, and the recognition test.

Manages Complexity¶

Without the abstraction, an analyst must reason directly over many local details: the carrier roles, admissibility assumptions, competing conventions, derived invariants, boundary cases, and proof or validation obligations specific to Cyclotomic polynomial. Cyclotomic polynomial compresses them into the roles in the structural signature. That compression permits comparison across instances without erasing the variables that determine validity. It also exposes which details may be varied safely and which are constitutive.

Abstract Reasoning¶

Identify the carrier. State what the elements, states, objects, or observations are: the typed algebra carrier, including objects, relations, parameters, conventions, evidence, boundaries, and comparison targets. Reject examples whose alleged carrier belongs to a different problem. 2. Lock the constitutive rule. Express the positive integer n, primitive nth roots and coprimality index, product definition, monicity and integer coefficients, irreducibility over rationals, degree Euler phi(n), factorization of x^n minus one, Möbius product and representative examples are explicit independently of one notation or implementation.

Knowledge Transfer¶

Knowledge transfers strongly among subfields of algebra because they reuse the typed algebra carrier, including objects, relations, parameters, conventions, evidence, boundaries, and comparison targets, Primitive roots are selected from all nth roots of unity and multiplied into a monic polynomial; Möbius inversion or recursive factorization of x^n−1 yields integer coefficients., and type the carrier, state every parameter and convention in the definition, test that the positive integer n, primitive nth roots and coprimality index, product definition, monicity and integer coefficients, irreducibility over rationals, degree Euler phi(n), factorization of x^n minus one, Möbius product and representative examples are explicit, compare the nearest accepted identity, and report counterexamples, uncertainty, and limiting cases.

Relationships to Other Abstractions¶

Current abstraction Cyclotomic polynomial Domain-specific

Parents (1) — more general patterns this builds on

Cyclotomic polynomial is a kind of Classification Prime

The proposed strict upward parent is prime:classification.

Hierarchy path (1) — routes to 1 parentless root

Cyclotomic polynomial → Classification

Neighborhood in Abstraction Space¶

Cyclotomic polynomial sits in a crowded region of the domain-specific corpus (30^th percentile for distinctiveness): several abstractions share nearly its structure, so a description that fits it tends to fit its neighbors too.

Family — Polynomial Algebra & Field Structure (25 abstractions)

Nearest neighbors

Computed from structural-signature embeddings · 2026-09-08