Euclidean rings: we call a ring "Euclidean" if division with a remainder is possible in the ring. In integer numbers we can divide \(a\) by \(b\) with a remainder if there exist unique \(r\) and \(q\) such that \(r <b\) and \(a=bq+r\), now for Gaussian numbers we need to define what does "\(<\)" mean and what does "unique".
Prove that the division with remainder using the complex norm \(|x+yi| = (x+yi)(x-yi)\) is well defined. Namely for any \(a\) and \(b\) there exist unique (up to multiplicatively invertible element) Gaussian numbers \(q\) and \(r\) with \(|r| <|b|\).
As a corollary (you don’t have to prove but you can use it in later problems) any Gaussian number has a unique (up to \(1,-1,i,-i\)) prime decomposition in \(\mathbb{Z}[i]\).