## 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 $\mathbb{Z}[i]$.