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

  • Thread starter Firepanda
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[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:

Answers and Replies

  • #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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