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 $1$ if and only if it equals $1$.

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Hence the only factor is $x - 1$.

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