First Cyclotomic Polynomial
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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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