Octopedia The Cyclotomic Polynomial Theorem
For any \(0 < n \in \mathbb{N}\), there are exactly one irreducible factor \(\Phi_{n}\) of \(1 - \alpha^{n}\) which is not a factor of any \(1 - \alpha^{k}\) where \(k\) is some number less than \(n\) dividing \(n\) (\(k\ \vert\ n\), \(k \lt n\))
\[\Phi_{n} \equiv n^{th} \text{ cyclotomic polynumber }\]The Cyclotomic factor formula
The product of all the natural numbers \(d\) which divide \(n\) of \(\Phi_{d}\)
\[\text{For } 0 < n \in \mathbb{N} \\ 1 - \alpha^{n} = \prod_{d|n}\Phi_{d}\]Examples
\[\displaylines{ \begin{align*} 1 - \alpha^{1} &= (1 - \alpha)\\ 1 - \alpha^{2} &= (1 - \alpha) (1 + \alpha)\\ 1 - \alpha^{3} &= (1 - \alpha)(1 + \alpha + \alpha^{2})\\ 1 - \alpha^{4} &= (1 - \alpha)(1 + \alpha)(1 + \alpha^{2})\\ 1 - \alpha^{5} &= (1 - \alpha)(1 + \alpha + \alpha^{2} + \alpha^{3} + \alpha^{4})\\ 1 - \alpha^{6} &= (1 - \alpha)(1 + \alpha)(1 + \alpha + \alpha^{2})(1 - \alpha + \alpha^{2})\\ 1 - \alpha^{7} &= (1 - \alpha)(1 + \alpha + \alpha^{2} + \alpha^{3} + \alpha^{4} + \alpha^{5} + \alpha^{6})\\ 1 - \alpha^{8} &= (1 - \alpha)(1 + \alpha)(1 + \alpha^{2})(1 + \alpha^{4})\\ 1 - \alpha^{9} &= (1 - \alpha)(1 + \alpha + \alpha^{2})(1 + \alpha^{3} + \alpha^{6})\\ 1 - \alpha^{10} &= (1 - \alpha)(1 + \alpha)(1 + \alpha + \alpha^{2} + \alpha^{3} + \alpha^{4})(1 - \alpha - \alpha^{2} - \alpha^{3} - \alpha^{4})\\ 1 - \alpha^{11} &= (1 - \alpha)(1 + \alpha + \alpha^{2} + \alpha^{3} + \alpha^{4} + \alpha^{5} + \alpha^{6} + \alpha^{7} + \alpha^{8} + \alpha^{9} + \alpha^{10})\\ 1 - \alpha^{12} &= (1 - \alpha)(1 + \alpha)(1 + \alpha + \alpha^{2})(1 + \alpha^{2})(1 - \alpha + \alpha^{2})(1 - \alpha^{2} + \alpha^{4})\\ 1 - \alpha^{13} &= (1 - \alpha)(1 + \alpha + \alpha^{2} + \dots + \alpha^{12})\\ 1 - \alpha^{14} &= (1 - \alpha)(1 + \alpha)(1 + \alpha + \alpha^{2} + \dots + \alpha^{6})(1 + \alpha + \alpha^{2} + \alpha^{3} + \alpha^{4} - \alpha^{5} + \alpha^{6})\\ 1 - \alpha^{15} &= (1 - \alpha)(1 + \alpha + \alpha^{2})(1 + \alpha + \alpha^{2} + \alpha^{3} + \alpha^{4})(1 - \alpha + \alpha^{3} - \alpha^{4} + \alpha^{5} - \alpha^{7} + \alpha^{8})\\ 1 - \alpha^{16} &= (1 - \alpha)(1 + \alpha)(1 + \alpha^{2})(1 + \alpha^{4})(1 + \alpha^{8})\\ \end{align*} }\]Prop
If \(p\) is a prime, then:
\[\Phi_{p} = 1 + \alpha + \alpha^{2} + \dots + \alpha^{p - 1}\]Examples
\[\displaylines{ \begin{align*} \Phi_{1} &= (1 - \alpha)\\ \Phi_{2} &= (1 + \alpha)\\ \Phi_{3} &= (1 + \alpha + \alpha^2)\\ \Phi_{4} &= (1 + \alpha^{2})\\ \Phi_{5} &= (1 + \alpha + \alpha^{2} + \alpha^{3} + \alpha^{4})\\ \Phi_{6} &= (1 - \alpha + \alpha^{2})\\ \Phi_{7} &= (1 + \alpha + \alpha^{2} + \alpha^{3} + \alpha^{4} + \alpha^{5} + \alpha^{6})\\ \Phi_{8} &= (1 + \alpha^{4})\\ \Phi_{9} &= (1 + \alpha^{3} + \alpha^{6})\\ \Phi_{10} &= (1 - \alpha + \alpha^{2} + \alpha^{3} + \alpha^{4})\\ \Phi_{11} &= (1 + \alpha + \alpha^{2} + \alpha^{3} + \alpha^{4} + \alpha^{5} + \alpha^{6} + \alpha^{7} + \alpha^{8} + \alpha^{9} + \alpha^{10})\\ \Phi_{12} &= (1 - \alpha^{2} + \alpha^{4})\\ \Phi_{13} &= (1 + \alpha + \alpha^{2} + \alpha^{3} + \alpha^{4} + \alpha^{5} + \alpha^{6} + \alpha^{7} + \alpha^{8} + \alpha^{9} + \alpha^{10} + \alpha^{11} + \alpha^{12})\\ \Phi_{14} &= (1 - \alpha + \alpha^{2} - \alpha^{3} + \alpha^{4} - \alpha^{5} + \alpha^{6})\\ \Phi_{15} &= (1 - \alpha + \alpha^{3} - \alpha^{4} + \alpha^{5} - \alpha^{7} + \alpha^{8})\\ \Phi_{16} &= (1 + \alpha^{8})\\ \end{align*} }\]Eisensteins Criterion
The cyclotomic polynomial with form:
\[1 - \alpha^{p} = (1 - \alpha)(1 + \alpha + \alpha^{2} + \dots + \alpha^{p - 1})\]is irreducible by Eisenstein’s criterion which is commonly known as Schönemann–Eisenstein theorem after Gotthold Eisenstein and Theodor Schönemann