Question on primitive roots of unity

Question

Let $p$ be an odd prime and $\omega$ be a primitive $p$^th root of unity. The question is to prove that:

$$(1-\omega)(1-\omega^2) \cdots (1-\omega^{p-1})=p$$

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What I have done so far is:

I can see that this is true for $p=3$

$$(1-\omega)(1-\omega^2)=1-(\omega+\omega^2)+w^3=1-(-1)+1=3=p$$

I am not able to prove this in general....

$$(1-\omega)(1-\omega^2) \cdots (1-\omega^{p-1})=1-(\omega+\omega^2+\cdots+\omega^{p-1})+????+\omega^{\frac{p(p-1)}{2}}$$

I do not have any idea what that $????$ could be but all I can say is:

• $(\omega+\omega^2+\cdots +\omega^{p-1})=-1$
• $\omega^{\frac{p(p-1)}{2}}=1$

So, $(1-\omega)(1-\omega^2)\cdots (1-\omega^{p-1})=3+????$

I am not able to do more than this.

There could be some (hard) way of doing it by hand multiplying all those things but I am looking for a more theoretical idea.

Please help me to clear this up.

Answer 1

Recall the formula for the summation $1+x+x^2+\cdots+x^n = {x^{n+1}-1 \over x-1}$ when $x\neq 1$.

In particular, this shows that $f(x)=x^{p}-1 =(1+x+x^2+\cdots+x^{p-1})(x-1)$.

Since $\omega$ is a root of unity, we have $f(\omega^k) = 0$ for $k=0,...,p-1$.

Since $\omega$ is primitive, $\omega^0,...,\omega^{p-1}$ are distinct and so $f(x) = (x-1)\cdots (x-\omega^{p-1}) $.

Hence $1+x+x^2+\cdots+x^{p-1} = (x-\omega^1)\cdots (x-\omega^{p-1}) $.

Answer 2

Hint: what does $x^n-1$ look like factored?

Answer 3

Let $f(x)=x^p-1$ and $g(x)=f(1-x)$. Then $g(1-\omega^k)=f(\omega^k)=0$ for all $k$.

$(1-\omega)(1-\omega^2)\cdots (1-\omega^{p-1})$ is thus the product of all roots of $g$ except $0$. So, expand $g$, factor $x$, and look at the independent term.