A fact about divisibility in ZZ[i]

Prove that if a and b are relatively prime in \mathbb{Z}, then they are relatively prime in \mathbb{Z}[i].


Suppose a,b \in \mathbb{Z} are relatively prime. Then there exist x,y \in \mathbb{Z} such that ax + by = 1. Suppose now that \gamma is a common divisor of a and b; say a = \gamma \xi and b = \gamma \eta. Then \gamma(x \xi + y \eta) = 1 in \mathbb{Z}[i], so that \gamma is a unit. In particular, a and b are relatively prime in \mathbb{Z}[i].

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