Question regarding field

Question

For any positive integer $k$ and prime $p$ find necessary and sufficient condition for $Z_p[\sqrt k] =\{a+b{\sqrt k}\mid a,b \in Z_p\}$ to be a field. Any help would be appreciated. While I am getting for any prime $p$ and $k.$ I don't think it's right. Thanks.

Answer 1

For me, the correct interpretation of $\mathbb Z_p[\sqrt k]$ is $ \mathbb Z_p[X]/(X^2-k) $.

With this interpretation, we have $\mathbb Z_p[\sqrt k]$ is a field iff $k$ is not a square mod $p$, because when $k \equiv a^2 \bmod p$ we have $$ \mathbb Z_p[X]/(X^2-k) = \mathbb Z_p[X]/(X^2-a^2) \cong \mathbb Z_p[X]/(X-a) \times \mathbb Z_p[X]/(X+a) \cong \mathbb Z_p \times \mathbb Z_p $$