Cyclotomic polynomial

Definition

Let $n$ be a natural number. The cyclotomic polynomial of degree $n$, denoted ${\mathrm{\Phi }}_n$, is defined in the following ways:

• It is the product, over all primitive $n^{th}$ roots $\zeta$ of $1$, of the linear polynomials $x-\zeta$.
• It is the minimal polynomial of any primitive $n^{th}$ root of unity.

Equivalence of definitions

The equivalence of definitions essentially follows by using definition (1) and showing, from that definition, that cyclotomic polynomials are irreducible.