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,