# Definition:Cyclotomic Polynomial

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## Definition

Let $n \ge 1$ be a natural number.

The **$n$th cyclotomic polynomial** is the polynomial

- $\ds \map {\Phi_n} x = \prod_\zeta \paren {x - \zeta}$

where the product runs over all primitive complex $n$th roots of unity, that is, those whose order is $n$.

From Condition for Complex Root of Unity to be Primitive it is seen that this can be expressed as:

- $\ds \map {\Phi_n} x = \prod_{\substack {1 \mathop \le k \mathop \le n \\ \gcd \set {k, n} = 1} } \paren {x - \map \exp {\frac {2 \pi i k} n} }$

## Examples

### First Cyclotomic Polynomial

The **first cyclotomic polynomial** is:

- $\map {\Phi_1} x = x - 1$

### Cyclotomic Polynomial of Prime Index

Let $p$ be a prime number.

The **$p$th cyclotomic polynomial** is:

- $\map {\Phi_p} x = x^{p - 1} + x^{p - 2} + \cdots + x + 1$

## Also see

- Results about
**cyclotomic polynomials**can be found here.

## Sources

- 2008: David Nelson:
*The Penguin Dictionary of Mathematics*(4th ed.) ... (previous) ... (next): Entry:**cyclotomic polynomial**