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Division algorithm and unique Gaussian integers

  1. Sep 26, 2009 #1
    1. The problem statement, all variables and given/known data
    Theorem
    Let [tex]\alpha\neq0[/tex] and [tex]\beta[/tex] be Gaussian integers. Then there are Gaussian integers [tex]\tau[/tex] and [tex]\rho[/tex] such that [tex]\beta=\tau\alpha+\rho[/tex] and [tex]N\left(\rho\right)<N\left(\alpha\right)[/tex]

    Problem
    Show that the Guassian integers [tex]\tau[/tex] and [tex]\rho[/tex] in the Theorem are unique if and only if [tex]\beta[/tex] is a multiple of [tex]\alpha[/tex]

    2. Relevant equations



    3. The attempt at a solution
    Can someone please give me a jumping off point for this question because I am not sure how to proceed? I used the theorem in previous problems to actually solve for [tex]\tau[/tex] and [tex]\rho[/tex] given [tex]\beta[/tex] and [tex]\alpha[/tex], but I am not sure how to go about this problem. Thanks
     
  2. jcsd
  3. Oct 7, 2009 #2
    I did some poking around trying to find hints but can not seem to find any beyond proving uniqueness for the division algorith for real numbers. I can not seem to find a way to apply a similar method here. Help please.
     
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