I am working through a exercise that asks to find an extension of Fermat's little theorem in the ring $\mathbb{Z}[\sqrt{-7}]$.

Basically it want an expression for $\alpha^p$ where $p$ is a prime greater than $7$ and $\alpha \in \mathbb{Z}[\sqrt{-7}]$.

I got an answer by using quadratic reciprocity as:

$$\alpha^p \equiv \begin{Bmatrix}\alpha ~\text{if}~ p \equiv 1,2,-3 \mod 7 \\ \alpha^*: ~\text{if}~ p \equiv -1,-2,3 \mod 7\end{Bmatrix} \mod p$$

I wondered if there is a way I could check whether this answer is correct? Does anyone know if this is true?



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