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Homework Help: How to interpret quotient rings of gaussian integers

  1. Feb 24, 2013 #1
    1. The problem statement, all variables and given/known data

    This is just a small part of a larger question and is quite simple really. It's just that I want to confirm my understanding before moving on.

    What are some of the elements of [itex]Z/I[/itex] where I is an ideal generated by a non-zero non-unit integer. For the sake of argument, lets take I=<3>.

    2. Relevant equations

    3. The attempt at a solution
    Representatives from one coset would be the following...
    (8+4i)/<3>=(5+i)/<3>=(2+i)[itex]\in Z/<3>[/itex]

    Representatives from another coset would be the following...
    (7+5i)/<3>=(4+2i)/<3>=(1+2i)[itex]\in Z/<3>[/itex]
  2. jcsd
  3. Feb 24, 2013 #2
    At first glance I think it would be [itex]\mathbb{Z}_3[/itex]. Can you try to prove this?
  4. Feb 24, 2013 #3

    I found the answer in that link on page 17. You are correct the elements are isomorphic to the integers mod 3 adjoin i. I just didn't understand that the quotient ring I was trying to produce the elements of was the (integers adjoin i) mod the generator of the ideal.
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