1. ## is ideal maximal?

Is the ideal $(1+3\sqrt{2}, 3+11\sqrt{2}, -1-11\sqrt{2})$ maximal in $\mathbb{Z}[\sqrt{2}]$?

I find that $gcd(1+3\sqrt{2}, 3+11\sqrt{2}, -1-11\sqrt{2})=(-1-\sqrt{2})$, which is irreducible in $\mathbb{Z}[\sqrt{2}].
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Does this mean that the ideal is maximal?
I guess the question is equivalent to the question is $\mathbb{Z}[\sqrt{2}]$ Euclidean doman, but I don't know is it.

Is there a rule when $\mathbb{Z}[d]$ is Euclidean domain, depending on $d$?

Thank you.

2. Originally Posted by georgel
Is the ideal $(1+3\sqrt{2}, 3+11\sqrt{2}, -1-11\sqrt{2})$ maximal in $\mathbb{Z}[\sqrt{2}]$?

I find that $gcd(1+3\sqrt{2}, 3+11\sqrt{2}, -1-11\sqrt{2})=(-1-\sqrt{2})$, which is irreducible in $\mathbb{Z}[\sqrt{2}].
$

Does this mean that the ideal is maximal?
Yes it is maximal if it is generated by an irreducible element.

I guess the question is equivalent to the question is $\mathbb{Z}[\sqrt{2}]$ Euclidean doman, but I don't know is it.
Yes, under the norm $a+b\sqrt{2} \mapsto |a^2 - 2b^2|$.

Is there a rule when $\mathbb{Z}[d]$ is Euclidean domain, depending on $d$?
This question is answered by the number theory book of Hardy and Wright.