Why the sequence of coefficients of a cyclotomic polynomial is a palindrome (idea)

Read cyclotomic polynomial first. Here's why such a polynomial has a sequence of coefficients which reads the same in both directions, maybe with a change of signs. Suppose t is a primitive n'th root of unity. Then so is its conjugate. But since |t|=1, 1/t is the conjugate of t.

Now, the n'th cyclotomic polynomial is simply the minimal polynomial (over Z) for t, hence also for its conjugate 1/t. Writing out the polynomial as

p(x) = a_k x^k + ... + a₀,

we see that

0 = a_k t^k + ... + a₀

and

0 = a_k 1/t^k + ... + a₀.

Multiplying the last equation by t^k, we get

0 = a₀ t^k + ... + a_k.

So if q(x) is the polynomial with reverse coefficient sequence compared to p(x), we have that q(t)=0 and has the same degree as p(x), and is therefore a multiple of it. But a₀ is the product of all primitive n'th roots of unity, hence is either +1 or -1. Thus either p(x)=q(x) or p(x)=-q(x), as required...