First Cyclotomic Polynomial

Theorem
The first cyclotomic polynomial is:
 * $\Phi_1 \left({x}\right) = x - 1$

Proof
By definition:
 * $\displaystyle \Phi_1 \left({x}\right) = \prod_\zeta \left({x - \zeta}\right)$

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 $x - 1$.