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Math Help - Algebraic number theory.

  1. #1
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    Algebraic number theory.

    Let I be an ideal generated by 2 and 1+\sqrt{-3} in the ring \mathbb Z[{\sqrt{-3}}].
    Show that I is different (2).
    And I^2 =2I.
    Show that ideals in \mathbb Z[{\sqrt{-3}}]. do not factor uniquely into prime ideals.
    Show that I is a unique prime ideal contaning (2).
    Conclude (2) is not a product of prime ideals.
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  2. #2
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    Quote Originally Posted by peteryellow View Post
    Let I be an ideal generated by 2 and 1+\sqrt{-3} in the ring \mathbb Z[{\sqrt{-3}}].
    Show that I is different (2).
    And I^2 =2I.
    Show that ideals in \mathbb Z[{\sqrt{-3}}]. do not factor uniquely into prime ideals.
    Show that I is a unique prime ideal contaning (2).
    Conclude (2) is not a product of prime ideals.
    i normally don't answer questions with so many parts and no work done by the OP, because that would be a crime! to get you started:

    most parts of the problem are easily proved. what you need to know here is that I is actually a maximal ideal of your ring. also 2I \subseteq I^2 is

    obvious because 2 \in I. to prove that I^2 \subseteq 2I, you only need to show that ab \in 2I, for all a,b \in I. for that you will need this fact that

    (1+\sqrt{-3})^2=-2 + 2 \sqrt{-3}=2(1+\sqrt{-3} - 2) \in 2I.
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