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Math Help - order of an ideal ring

  1. #1
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    order of an ideal ring

    How many elements are in Z[i]/<3+i>. Explain your answer.


    Can someone please show me how to find the order of an ideal complex ring?

    For this problem, I guess there are 10 elements in the ring.

    Thank you.
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  2. #2
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    Quote Originally Posted by john_n82 View Post
    How many elements are in Z[i]/<3+i>. Explain your answer.


    Can someone please show me how to find the order of an ideal complex ring?

    For this problem, I guess there are 10 elements in the ring.

    Thank you.
    in general the ring \frac{\mathbb{Z}[i]}{<a+bi>} has a^2 + b^2 elements. to see this start with the fact that \mathbb{Z}[i] is a euclidean domain. so for any x \in \mathbb{Z}[i], there exist s, r \in \mathbb{Z}[i] such that x=s(a+bi) + r

    and either r=0 or |r|^2 < a^2 +b^2.
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  3. #3
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    Quote Originally Posted by NonCommAlg View Post
    in general the ring \frac{\mathbb{Z}[i]}{<a+bi>} has a^2 + b^2 elements. to see this start with the fact that \mathbb{Z}[i] is a euclidean domain. so for any x \in \mathbb{Z}[i], there exist s, r \in \mathbb{Z}[i] such that x=s(a+bi) + r

    and either r=0 or |r|^2 < a^2 +b^2.
    (First a+bi \not = 0 but that is a trivial thing).

    What you wrote is correct, however, I do not see how it follows from 0\leq |r|^2 < a^2+b^2.

    Because if you take a+bi=1+i, for example, then there is non-uniqueness for the division algrorithm.
    Notice if r=n+mi then we want 0\leq n^2+m^2 < 2. There are 5 choices for n,m\in \mathbb{Z} but there are only |1+i|^2 = 2 congruence classes.
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