Programming ECC

Suppose ${𝔽}_q[\alpha ]$ is a quadratic extension of $F_q$ where ${\alpha }^2=\delta$ for some $\delta \in {𝔽}_q$. Then let $x=a+\alpha b$ be some $r$th root of unity in ${𝔽}_q$ for some $r$ dividing $q+1$. (Note $q+1={\Phi }_2(q)$.)

Then $x^{q+1}=1$, thus $$1=a^{q+1}+{\alpha }^{q+1}{b}^{q+1}=a^2+({\alpha }^q)\alpha b^2=a^2-\delta b^2$$ since ${\alpha }^q=-\alpha$.