The question is “Establish an analogue of Fermat’s Little Theorem for the ring $\mathbb{Z} [\sqrt{-2}]$.”
I know how to do this for the cases where $\mathbb{Z} [\sqrt{3}]$ and $\mathbb{Z} [\sqrt{5}]$ by letting $\alpha \in \mathbb{Z} [\sqrt{3}]$ so there exists integers $a$ and $b$ s.t $\alpha^{p}=(a+b\sqrt{3})^p$ where $p$ is prime. Then by binomial expanding and using Fermat’s Little Theorem you get $\alpha^{p}=a+b(\sqrt{3})^{p}$ mod$p$. Then you can use Euler’s criterion and the definition of the Legendre symbol for $(\frac{3}{p})$ to establish an analogue. However if I follow this through for the ring $\mathbb{Z} [\sqrt{-2}]$ I get the Legendre Symbol $(\frac{-2}{p})$and I have no idea how to calculate it. Am I doing something wrong or is this the correct method and am I just stumbling at the final part?