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Math Help - is ideal maximal?

  1. #1
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    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}].<br />
    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.
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  2. #2
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    Quote Originally Posted by georgel View Post
    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}].<br />
    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.
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