Which quotient rings of $\Bbb Z [\sqrt{3}]$ are fields?

Question

Please help. I have this problem as my homework but I couldn't solve it.

Determine all ideals $I$ of $\Bbb Z [\sqrt {3}]$ such that the quotient ring $\Bbb Z [\sqrt {3}] / I$ is a field. Is $\Bbb Z [\sqrt {3}]$ an euclidean domain?

Answer 1

Since $\Bbb Z[\sqrt 3]\cong \frac{\Bbb Z[x]}{(x^2-3)}$, you will just have to determine the maximal ideals of $\Bbb Z[x]$ containing $(x^2-3)$. Fortunately, these will be among the prime ideals of $\Bbb Z[x]$, which are well-known.

Hint for the Euclidean domain part: Try the norm $N(\alpha+\sqrt{3}\beta)=|\alpha^2-3\beta^2|$