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Math Help - Principal Ideal

  1. #1
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    Principal Ideal

    How do I show the ideal $I = \left( {1 - \sqrt { - 5} } \right)$ is principal in ${Z}\left[ {\sqrt { - 5} } \right$

    Thanks!
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  2. #2
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    Quote Originally Posted by orbit View Post
    How do I show the ideal $I = \left( {1 - \sqrt { - 5} } \right)$ is principal in ${Z}\left[ {\sqrt { - 5} } \right$

    Thanks!


    You should really try to think over your questions before you send them: in ANY ring, the ideal <x> is principal by DEFINITION (read it)

    Tonio
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  3. #3
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    Yeah.... are you sure you didn't mean something else? I mean, in rings like \mathbb{Z}[\sqrt{-5}], there CAN be interesting questions of this sort in such rings.... but this one clearly is not.
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  4. #4
    MHF Contributor FernandoRevilla's Avatar
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    Quote Originally Posted by topspin1617 View Post
    Yeah.... are you sure you didn't mean something else? I mean, in rings like \mathbb{Z}[\sqrt{-5}], there CAN be interesting questions

    Right, for example 3,2+\sqrt{-5},2-\sqrt{-5} are irreducible elements in \mathbb{Z}[\sqrt{-5}] and 9=3\cdot 3=(2+\sqrt{-5})\cdot (2-\sqrt{-5}) so, \mathbb{Z}[\sqrt{-5}] is not a factorial ring. This example provided by Dedekind has had great importance in the foundation of ideals theory to supply the factorization of elements in not factorial rings.
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