User:Ga ohoyt/Cyclotomic polynomials

Cyclotomic polynomials are the irreducible factors of when the polynomial coefficients are restricted to the field of rational numbers. For example

z⁶−1 = (z−1) (z+1) (z²+z+1) (z²−z+1).

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

z⁶−1 = .

The zeroes of the polynomial

are precisely the n^th roots of unity, each with multiplicity 1.

The n^th cyclotomic polynomial is defined by the fact that its zeros are precisely the primitive nth roots of unity, each with multiplicity 1:

where z₁,...,z_φ(n) are the primitive n^th roots of unity, and is Euler's totient function. The polynomial has integer coefficients and is an irreducible polynomial over the rational numbers (i.e., cannot be written as a product of two positive-degree polynomials with rational coefficients). (The case of prime n, which is easier than the general assertion, follows from Eisenstein's criterion.)

Every n^th root of unity is a primitive d^th root of unity for exactly one positive divisor d of n. This implies that

This formula represents the factorization of the polynomial zⁿ - 1 into irreducible factors.

z¹−1 = z−1

z²−1 = (z−1)(z+1)

z³−1 = (z−1)(z²+z+1)

z⁴−1 = (z−1)(z+1)(z²+1)

z⁵−1 = (z−1)(z⁴+z³+z²+z+1)

z⁶−1 = (z−1)(z+1)(z²+z+1)(z²−z+1)

z⁷−1 = (z−1)(z⁶+z⁵+z⁴+z³+z²+z+1)

Applying Möbius inversion to the formula gives

where μ is the Möbius function.

So the first few cyclotomic polynomials are

Φ₁(z) = z−1

Φ₂(z) = (z²−1)(z−1)⁻¹ = z+1

Φ₃(z) = (z³−1)(z−1)⁻¹ = z²+z+1

Φ₄(z) = (z⁴−1)(z²−1)⁻¹ = z²+1

Φ₅(z) = (z⁵−1)(z−1)⁻¹ = z⁴+z³+z²+z+1

Φ₆(z) = (z⁶−1)(z³−1)⁻¹(z²−1)⁻¹(z−1) = z²−z+1

Φ₇(z) = (z⁷−1)(z−1)⁻¹ = z⁶+z⁵+z⁴+z³+z²+z+1

If p is a prime number, then all p^th roots of unity except 1 are primitive p^th roots, and we have

. Substituting for , this is a base z repunit. Thus a necessary (but not sufficient) condition for a repunit to be prime is that its length be prime.

Note that, contrary to first appearances, not all coefficients of all cyclotomic polynomials are 1, −1, or 0; the first polynomial where this occurs is , since 105=3×5×7 is the first product of three odd primes. Many restrictions are known about the values that cyclotomic polynomials can assume at integer values. For example, if is prime and then or .