# User:Ga ohoyt/Cyclotomic polynomials

**Cyclotomic** (or "circle dividing") **polynomials** are the irreducible factors of when the polynomial coefficients are restricted to the field of rational numbers. For example

- .

The cyclotomic polynomials are often written as . The polynomial has exactly as many of these factors as there are integer factors of n:

- .

## Contents

## Finding cyclotomic polynomials

### Recursive method

Cyclotomic polynomials are defined by the equation

- .

Solving for , one obtains

- .

One can then construct any polynomial recursively, given that :

- .

### Direct method

For larger values of n, it is easier to use the direct formula, based on the Möbius inversion of the above formula:

where μ is the Möbius function. For example, one would follow these steps for n = 60:

**1.**Factor n.

**2.**Determine the square-free divisors of n, and classify them according to whether their number of prime factors are even or odd.

- (Odd)
- (Even)
- (Even)
- (Even)
- (Odd)
- (Odd)
- (Odd)
- (Even, since 1 is not a prime)

**3.**Form the factors , putting the factors from the even class in the numerator and those from the odd class in the denominator.

- .

### Construction from complex roots

It is also possible to construct a cyclotomic polynomial from the complex roots of 1:

where is Euler's totient function. For example,

## Some special cases

When p and q are odd primes,