Definition:Cyclotomic Polynomial

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 $n$.

Also presented as

Based on Condition for Complex Root of Unity to be Primitive, the cyclotomic polynomial can also 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

• Cyclotomic Polynomial has Integer Coefficients

• Results about cyclotomic polynomials can be found here.

Sources

• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): cyclotomic polynomial
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): cyclotomic polynomial