Show that , where is a primitive 8th root of unity.

Proof. Now . So I have

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- March 26th 2008, 09:43 AMtttcomraderPrimitive root of unity problem
Show that , where is a primitive 8th root of unity.

Proof. Now . So I have - March 26th 2008, 10:08 AMtopsquark
- March 26th 2008, 10:33 AMThePerfectHacker
- March 26th 2008, 11:34 AMtopsquark
- March 28th 2008, 06:38 AMawkward
Well, actually, there are 4 primitive 8th roots of unity ( ), so presumably your proof should work for any one of them.

Let be a primitive 8th root of unity. Note that

so .

(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, .

And

so , hence .