1. ## is ideal maximal?

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

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

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

Thank you.

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

I find that $\displaystyle gcd(1+3\sqrt{2}, 3+11\sqrt{2}, -1-11\sqrt{2})=(-1-\sqrt{2})$, which is irreducible in $\displaystyle \mathbb{Z}[\sqrt{2}].$
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 $\displaystyle \mathbb{Z}[\sqrt{2}]$ Euclidean doman, but I don't know is it.
Yes, under the norm $\displaystyle a+b\sqrt{2} \mapsto |a^2 - 2b^2|$.

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