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Problem 21 (Adjoining The Square Root Of A Positive Integer To $\mathbb{Z}_p$)
Show that $\mathbb{Z}_7[\sqrt{3}] = \{a+b\sqrt{3}\mid a,b\in \mathbb{Z}_7\} $ is a field. Given a prime $p$ and positive integer $k$, determine conditions that are both necessary and sufficient for the ring $\mathbb{Z}_p[\sqrt{k}] = \{a+b\sqrt{k}\mid a,b\in \mathbb{Z}_p\}$ to be a field.
Click to see a hint.
Why is it enough to just determine conditions that cause $\mathbb{Z}_p[\sqrt{k}]$ to be an integral domain? You should be able to use systems of equations to reduce the problem to solving $$ \begin{bmatrix}a&-bk\\b&a\end{bmatrix} \begin{bmatrix}c\\d\end{bmatrix} = \begin{bmatrix}0\\0\end{bmatrix}. $$ When is the matrix invertible in $\mathbb{Z}_p$? This should give you your necessary and sufficient conditions.
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