# Cyclotomic

Cyclotomic[n,x]

gives the n cyclotomic polynomial in x.

# Details • The cyclotomic polynomial of order is defined to be , where the product runs over integers less than that are relatively prime to .

# Examples

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## Basic Examples(1)

The roots are the primitive 5 roots of :

## Applications(6)

Values of successive cyclotomic polynomials at 1:

Calculate unique primes for which the decimal expansion of has a unique period:

Plot cyclotomic polynomials:

Plot arguments of roots of Cyclotomic:

Plot the degree and number of terms of cyclotomic polynomials:

## Properties & Relations(7)

Factor a cyclotomic polynomial over an extension field:

Generate cyclotomic polynomials from a definition:

Use an alternative definition, valid for :

Form products of cyclotomic polynomials:

Plot the Riemann surface of an inverse of a cyclotomic polynomial over the complex plane:

Plot the complex roots of successive derivatives of the 50 cyclotomic polynomial:

## Neat Examples(2)

The first cyclotomic polynomial with a coefficient other than 0, ±1:

Nonzero coefficients of successive cyclotomic polynomials: