I am working in the ring Z[sqrt(-5)]. I have shown that 1 is a gcd(x, y) where x=3 and y=2+sqrt(-5). I would like to show however that 1 cannot be expressed as ax+by where a, b are in Z[sqrt(-5)]. I have tried letting a and b be arbitrary elements of the ring and rearranging ax+by=1 to try to find a contradiction but this has not worked. Can anybody help? Thank you!
2026-05-05 08:00:19.1777968019
1 is gcd(x,y) but 1 cannot be expressed as ax+by where a,b,x,y are in Z[sqrt(-5)].
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Consider integers $m,n,p,q$ such that $$a=m+n\sqrt{-5}$$ $$b=p+q\sqrt{-5}$$
Then $3m+3n\sqrt{-5}+2p-5q+\sqrt{-5}(p+2q)=1$. That is: $$3m+2p-5q=1$$ and $$3n+p+2q=0$$ Now, solve for $p$ in the latter equation and substitute in the former to get $$3m-6n-4q-5q=1$$ Is that possible?