gives the n^(th) cyclotomic polynomial in x.


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


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Basic Examples  (1)

The roots are the primitive 5^(th) roots of :

Scope  (1)

TraditionalForm formatting:

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^(th) cyclotomic polynomial:

Neat Examples  (2)

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

Nonzero coefficients of successive cyclotomic polynomials:

Wolfram Research (1988), Cyclotomic, Wolfram Language function,


Wolfram Research (1988), Cyclotomic, Wolfram Language function,


@misc{reference.wolfram_2020_cyclotomic, author="Wolfram Research", title="{Cyclotomic}", year="1988", howpublished="\url{}", note=[Accessed: 15-January-2021 ]}


@online{reference.wolfram_2020_cyclotomic, organization={Wolfram Research}, title={Cyclotomic}, year={1988}, url={}, note=[Accessed: 15-January-2021 ]}


Wolfram Language. 1988. "Cyclotomic." Wolfram Language & System Documentation Center. Wolfram Research.


Wolfram Language. (1988). Cyclotomic. Wolfram Language & System Documentation Center. Retrieved from