Definitions

$⟨$ Definition A-8.1 $⟩$ Cyclotomic Polynomial

The $𝐧$ -th Cyclotomic Polynomial: is a polynomial whose roots are the primitive $n$ -th root of unity, that is:

Φ_{n} (x) = \prod_{ζ \in P (n)} (x - ζ) = \prod_{\begin{array}{c} 0 \leq k \leq n - 1, \\ gcd (k, n) = 1 \end{array}} (x - ω^{k}) ......, where ω = e^{2 𝜋𝑖 ∕ n}

Remember the Euler’s formula: $e^{2 𝑘𝜋𝑖 ∕ n} = \cos (\frac{2 𝑘𝜋}{n}) + i \cdot \sin (\frac{2 𝑘𝜋}{n})$

A few pre-computed cyclotomic polynomials are as follows:

$Φ_{1} (x) = x - 1$
$Φ_{2} (x) = x + 1$
$Φ_{3} (x) = x^{2} + x + 1$
$Φ_{4} (x) = x^{2} + 1$
$Φ_{5} (x) = x^{4} + x^{3} + x^{2} + x + 1$
$Φ_{6} (x) = x^{2} - x + 1$
$Φ_{7} (x) = x^{6} + x^{5} + x^{4} + x^{3} + x^{2} + 1$
$Φ_{8} (x) = x^{4} + 1$
$Φ_{9} (x) = x^{6} + x^{3} + 1$
$Φ_{10} (x) = x^{4} - x^{3} + x^{2} - x + 1$

= (x - (\cos (\frac{π}{2}) + i \cdot \sin (\frac{π}{2}))) \cdot (x - (\cos (\frac{3 π}{2}) + i \cdot \sin (\frac{3 π}{2})))

A-8.1 Definitions