1. Not finding help here? Sign up for a free 30min 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!

Field of quotients

  1. Feb 1, 2008 #1
    1. The problem statement, all variables and given/known data
    Describe the field of quotients of the integral subdomain D = {n+mi|n,m in Z} of the field of complex numbers. "Describe" means give the elements of C that make the field of quotients of D in C.

    2. Relevant equations

    3. The attempt at a solution
    So any complex number that has the form (nn'+mm'+i(nm'+mn'))/(n'^2+m'^2) will be in the field...but how can I be more descriptive...
    Last edited: Feb 1, 2008
  2. jcsd
  3. Feb 1, 2008 #2


    User Avatar
    Science Advisor
    Homework Helper

    Do you mean a set generators of [tex]D[/tex]?
  4. Feb 1, 2008 #3
    I stated the problem the way it is stated in the book, but I guess a set of generators for Quot(D) would work.
  5. Feb 2, 2008 #4


    User Avatar
    Staff Emeritus
    Science Advisor

    The "field of quotients" of the sat {m + ni} where m and n are integers (the "Gaussian integers) is, by definition, the set of things of the form (m+ ni)/(a+ bi) where both a and b are also integers. Multiplying numerator and denominator of the fraction by a- bi will make the denominator an integer and give us something of the form (x/p)+ (y/p)i. Looks to me like the field of integers is the set of numbers of the form r+ si where r and s are rational numbers.
  6. Feb 2, 2008 #5
    That seems reasonable but I still need to prove that
    hits every number of the form r+si, where r and s are rational numbers...
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?