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) = ak xk + ... + a0,
we see that
0 = ak tk + ... + a0
and
0 = ak 1/tk + ... + a0.
Multiplying the last equation by
tk, we get
0 = a0 tk + ... + ak.
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...