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Find two elements in the domain $R := \{ x + 2y \sqrt {-1} \mid x,y \in \mathbb{Z} \}$ that do not have a gcd.


I have no idea how to start. But I know if we consider $R^\prime = \{ x + y \sqrt {-1} \mid x,y \in \mathbb{Z} \}$ then every two elements have a gcd. So there must be something wrong with the 2 here.

Any help is appreciated.

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1 Answer 1

up vote 3 down vote accepted

So there must be something wrong with the 2 here.

Right. Use that $2-2i$ and $2$ are irreducible in $\Bbb Z[\sqrt{-4}]$ but have common factor $2$ in $\Bbb Z[i]$.

So $(2+2i)(2-2i) = 8 = 2^3$ and both $2-2i$ and $2$ are irreducible (consider the multiplicative norm $N(a+b\sqrt{-4}) = a^2+4b^2$ for a proof). Hence $4-4i=2(2-2i)$ and $8$ do not have a greatest common divisor.

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Thank you so much! That helps a lot! –  user112564 Jan 31 at 7:20
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