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Math Help - The number of elements in Gaussian Factor ring

  1. #1
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    Lightbulb The number of elements in Gaussian Factor ring

    Hi Guys ;

    how many elements are in \mathbb Z_5[i]/<1+i> ?

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  2. #2
    MHF Contributor Swlabr's Avatar
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    Quote Originally Posted by leonardo1000 View Post
    Hi Guys ;

    how many elements are in \mathbb Z_5[i]/<1+i> ?


    How many elements are in \mathbb Z_5[i]? What about <1+i>? Clearly, \mathbb Z_5[i] has 25 elements, so <1+i> has either 1, 5 or 25 elements. It's not got only 1 element, obviously (as it contains 0). I'll leave the rest of that bit to you...and the solution to your problem is simply |\mathbb Z_5[i]|/|<1+i>| = 25/|<1+i>|.
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  3. #3
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    Quote Originally Posted by leonardo1000 View Post
    Hi Guys ;

    how many elements are in \mathbb Z_5[i]/<1+i> ?

    one element only! the reason is that in \mathbb{Z}_5[i] we have 1=(3+2i)(1+i) \in <1+i>. thus <1+i>=\mathbb{Z}_5[i].
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