## Bezout’s Identity for the Gaussian integers

Prove that if $\alpha,\beta \in \mathbb{Z}[i]$ are relatively prime, then there exist $\mu, \eta \in \mathbb{Z}[i]$ such that $\alpha\mu + \beta\eta = 1$.

We have seen that $\mathbb{Z}[i]$ is a Euclidean domain, and thus a principal ideal domain. If 1 is a greatest common divisor of $\alpha$ and $\beta$, then $(\alpha,\beta) = \mathbb{Z}[i]$. The conclusion follows.