First Cyclotomic Polynomial

Theorem
The first cyclotomic polynomial is:
 * $\map {\Phi_1} x = x - 1$

Proof
By definition:
 * $\ds \map {\Phi_1} x = \prod_\zeta \paren {x - \zeta}$

where the product runs over all primitive complex first roots of unity.

A root of unity has order $1$ it equals $1$.

Hence the only factor is $x - 1$.