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[Number Theory] Finding principal ideals in Z[√-6]

  1. Mar 27, 2012 #1
    [Number Theory] Find all the ideals with the element 6 in them in Z[√-5]

    Edited original question since I have now found the answer (I realise the title is inconsistent on the forum page), instead I am now trying to do part i) here

    immohj.png

    Is it possible to it this way:

    2hmzyqd.png

    Or is the structure of the question meaning my lecturer wants the actual ideals, and not just a general form? If so, how do i find which ideals contain 6.

    I know the general form is

    a = paqbrc

    Whre a is an element of {0,1,2}, and b,c are an element of {0,1}

    So there are 9 different combinations I think, do I have to check all 9? Or do all 9 have this property and there is no need to check?

    Thanks
     
    Last edited: Mar 27, 2012
  2. jcsd
  3. Mar 28, 2012 #2

    morphism

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    Hint: An ideal contains 6 iff it contains (6)=(2)(3) (principal ideals) iff it divides (2)(3).

    So factor (2) and (3) in Z[sqrt{-6}].
     
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