1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

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

    micromass

    User Avatar
    Staff Emeritus
    Science Advisor
    Education Advisor
    2016 Award

    At first glance I think it would be [itex]\mathbb{Z}_3[/itex]. Can you try to prove this?
     
  4. Feb 24, 2013 #3
    http://www.math.uconn.edu/~kconrad/blurbs/ugradnumthy/Zinotes.pdf

    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.
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?
Draft saved Draft deleted



Similar Discussions: How to interpret quotient rings of gaussian integers
  1. Quotient ring? (Replies: 6)

Loading...