# Greatest Common Divisor in a strange extension ring.

by tomtom690
Tags: gcd, irreducible, prime, ring, z[sqrt(-5)]
 P: 7 1. The problem statement, all variables and given/known data I need to show that two elements in $$\textbf{Z}$$[$$\sqrt{-5}$$] have gcd = 1. The elements are 3 and 2+$$\sqrt{-5}$$ 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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