Greatest Common Divisor in a strange extension ring.by tomtom690 Tags: gcd, irreducible, prime, ring, z[sqrt(5)] 

#1
Dec1410, 09:11 AM

P: 7

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. 


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