cyclotomic polynomial

• x-1
• x+1
• x²+x+1
• x²+1
• x⁴+x³+x²+x+1

These are the first 5 cyclotomic polynomials. The nth cyclotomic polynomial is
cyc_n(x)=(x-e₁)(x-e₂)...(x-e_r)
where e₁,e₂,...,e_r are the primitive complex nth roots of unity.

It follows from the definition that the degree of cyc_n is phi(n), where phi is the Euler Phi function.

In the case of a prime number p we get
cyc_p(x)=(x^p-1)/(x-1)=x^p-1+x^p-2+...+x+1
(because all the pth roots of unity are primitive except for 1).

The cyclotomic polynomials can be calculated recursively because of the formula
xⁿ-1=product over all d|n of cyc_d(x)

This formula follows because the roots of xⁿ-1 are exactly the nth roots of unity. By Lagrange's Theorem each nth root of unity is a primitive dth root of unity for some divisor d of n. On the other hand clearly any such primitive dth root of unity is an nth root of unity.

Let's do this for an example, n=6. The divisors of 6 are 1,2,3,6 so the formula tells us
x⁶-1=cyc₁(x)cyc₂(x)cyc₃(x)cyc₆(x)
=(x-1)(x+1)(x²+x+1)cyc₆(x)
=(x²-1)(x²+x+1)cyc₆(x)
=(x⁴+x³-x+1)cyc₆(x)

We obtain from this that cyc₆(x)=x²-x+1.

A couple more interesting properties of cyc_n(x)

• it has integer coefficients and it is monic (i.e. the top degree term is a power of x). This follows from the formula, by induction.
• it is irreducible as a polynomial in Q[x] (polynomials with rational coefficients). See irreducibility of cyclotomic polynomials.
• You might conjecture that its coefficients are always 1 or -1 but this is not true...

[ 1, 1, 1, 0, 0, -1, -1, -2, -1, -1, 0, 0, 1, 1, 1, 1, 1, 1, 0, 0, -1, 0, -1, 0, -1, 0, -1, 0, -1, 0, 0, 1, 1, 1, 1, 1, 1, 0, 0, -1, -1, -2, -1, -1, 0, 0, 1, 1, 1 ],

Instead, I give you the following trivial fact about cyclotomic polynomials. Their sequence of coefficients is a palindrome: it reads the same in both directions, possibly changing signs on all coefficients! This is less astounding when you think about it. Here's why the sequence of coefficients of a cyclotomic polynomial is a palindrome...