Let $q$ be a prime or primer power such that $\mathbb{F}_q$ is a finite field. Now consider the extension $\mathbb{F}_{q^m}$, which may be regarded as a vector space of dimension $m$ over $\mathbb{F}_q$.
Let $\beta_1,\beta_2\in\mathbb{F}_{q^m}$. Now I want to show that $$\beta_1 \beta_2^q-\beta_2\beta_1^q=\beta_1 \prod_{c\in \mathbb{F}_q} \left( \beta_2-c\beta_1\right). $$ I can see that we will get a term $\beta_1\beta_2^q$ and $\beta_2\beta_1^q$ by mutiplying all the $\beta_2$ terms and all $\beta_1$ terms respectively (for one $c$ we have $c=0$, so we will only get $\beta_1^q$), but what about all the cross terms? How do we know that they will cancel out?
Cheers.