Greatest Common Divisor in a strange extension ring.

  1. 1. The problem statement, all variables and given/known data
    I need to show that two elements in [tex]\textbf{Z}[/tex][[tex]\sqrt{-5}[/tex]] have gcd = 1.
    The elements are 3 and 2+[tex]\sqrt{-5}[/tex]

    2. Relevant equations

    3. The attempt at a solution
    My way of thinking was if I can show that both elements are irreducible, then they are both prime and hence have gcd of 1. I can show they are both irreducible, using the norm function - ie showing that if eg 3 = ab then either N(a) or N(b) is 1. This means that 3 is irreducible in this ring. I think.
    Can somebody tell me if this is correct please? Like I said, I'm almost there, just need to polish it off!
    Thanks in advance.
  2. jcsd
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