Product of elements of a finite field.

Question

I want to show that for a finite field $\mathbb{F}_{p^n}$ $$\prod_{\alpha \in \mathbb{F}^{\times}_{p^n}}\alpha = (-1)^{p^n}$$

I'm not sure how to proceed with this. Any help is appreciated.

Answer 1

If $\alpha \not= 1/\alpha$ then the contribution of those terms to the total product is $\alpha(1/\alpha) = 1$.

By the way, just write the right side as $-1$: if $p$ is odd then $(-1)^{p^n} = -1$, and if $p = 2$ then $(-1)^{p^n} = 1 = -1$ since the field has characteristic $2$. When $n=1$ this result isWilson's theorem, which says $(p-1)! \equiv -1 \bmod p$ for prime $p$. If you can prove Wilson's theorem then you should try to adapt the ideas in that argument. If you can't prove Wilson's theorem then you should understand a proof of that special case first.