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Math Help - Integer Proof

  1. #1
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    Integer Proof

    Let w be in Z(square root of -3) with w not equal to 0,1,-1. Prove w can be factored into irreducible elements of Z(square root of -3); that is, we can find irriducible elements p1,p2,...,pt with w=p1p2...pt
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  2. #2
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    Quote Originally Posted by teramaries View Post
    Let w be in Z(square root of -3) with w not equal to 0,1,-1. Prove w can be factored into irreducible elements of Z(square root of -3); that is, we can find irreducible elements p1,p2,...,pt with w=p1p2...pt
    The factorization is not unique - you have:
    (1+\sqrt3i)(1-\sqrt3i)=4=2.2.

    But I think this can be shown by induction on |w| (which is a positive integer). Simply by the argument, that it is either irreducible, or it can be written as a product of two elements with smaller absolute value.

    Of course, you'll have to show that all units in this ring have absolute value 1 first, but this is easy.
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