cyclotomic polynomial

分圆多项式

Number Theory · Undergraduate · Mathematics Resources editorial compilation (AI-assisted)

Other names / variants: cyclotomic polynomial of order n

Translation evidence: Literature usage

Meaning in context

For $n\ge1$, $\Phi_n(X)$ is the monic polynomial whose roots are the complex roots of unity of exact order $n$.

对正整数 $n$，$\Phi_n(X)$ 是以所有阶恰为 $n$ 的复单位根为根的首一多项式。

Example

$\Phi_6(X)=X^2-X+1$.

Usage and distinctions

$X^n-1$ includes roots of all orders dividing $n$; $\Phi_n$ includes only primitive $n$th roots.

Recommended reading in this subject · Preparation & skip guidance

Definition sources

1. 张神星 — 代数数论讲义 v2.5 (2024-01-09)，合肥工业大学作者讲义Section 1.2.4, printed page 11, product defining Phi_N and factorization of T^N-1; n=1,2 are immediate extensions · Checked 2026-10-14

Translation sources

1. 张神星 — 代数数论讲义 v2.5 (2024-01-09)，合肥工业大学作者讲义§1.2.4，printed page 11 分圆多项式 · Checked 2026-10-14

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