Cyclotomic Polynomial - Examples

Examples

If n is a prime number then

If n=2p where p is an odd prime number then

For n up to 10 we have:

For n up to 20, the cyclotomic polynomials not covered by above formulas are:

The case of 105 is interesting because it is the first integer that is the product of three distinct odd prime numbers and the 105th cyclotomic polynomial is the first one that has coefficients of magnitude greater than 1:

\begin{align} \Phi_{105}(x) = & \; x^{48} + x^{47} + x^{46} - x^{43} - x^{42} - 2 x^{41} - x^{40} - x^{39} + x^{36} + x^{35} + x^{34} \\ & {} + x^{33} + x^{32} + x^{31} - x^{28} - x^{26} - x^{24} - x^{22} - x^{20} + x^{17} + x^{16} + x^{15} \\ & {} + x^{14} + x^{13} + x^{12} - x^9 - x^8 - 2 x^7 - x^6 - x^5 + x^2 + x + 1 \end{align}

Read more about this topic:  Cyclotomic Polynomial

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