Show that , where is a primitive 8th root of unity.
Proof. Now . So I have
Let be a primitive 8th root of unity. Note that
(If we had any other of the factors = 0, say , then would be an 8th root of unity but not a primitive root. Or, to put it another way, is the 8th cyclotomic polynomial.)
Since is a field, .
so , hence .