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!

Compute the G.C.D of two Gaussian Integers

  1. Oct 22, 2015 #1
    1. The problem statement, all variables and given/known data
    Hello all I apologize for the triviality of this:
    Im new to this stuff (its easy but unfamiliar) I was wondering if someone could verify this:

    Find the G.C.D of [itex]a= 14+2i [/itex] and [itex]b=21+26i [/itex].

    [itex] a,b \in \mathbb{Z} [ i ] [/itex] - Gaussian Integers

    2. Relevant equations

    None

    3. The attempt at a solution

    Well, is it true that any common divisor must also divide the G.C.D of the norm's of [itex] a [/itex]and[itex] b [/itex]?

    If so then, [itex] norm(14+2i)=200 [/itex]
    [itex]norm(21+26i)=1117 [/itex]

    Well, since 1117 and 200 are co-prime, their greatest common divisor is one. Thus,

    Thus the G.C.D of a,b is a unit (1,-1,i,-i) in the ring.

    Thanks
     
    Last edited: Oct 22, 2015
  2. jcsd
  3. Oct 22, 2015 #2

    mfb

    User Avatar
    2016 Award

    Staff: Mentor

    Everything that divides (a+bi) also divides (a+bi)(a-bi), sure.
    In the integers. You'll have to show that this is true for Gaussian integer factors as well.
     
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: Compute the G.C.D of two Gaussian Integers
Loading...