is in fact unique and that it generates $H$, i.e. each $h=ng$ for some $n\in\mathbb{Z}$. Intuitively, you can visualize this as a ( parallelogramic ) lattice in $\mathbb{Z}^2$, or as an additive subgroup (in fact, an ideal ) $I=\left<\alpha+i\beta\right>$ in the ring of Gaussian integers . To make the formal proof,