Ring of Gaussian integers

Definition
The ring of Gaussian integers $$\mathbb{Z}[i]$$ is defined in the following ways:


 * 1) It is the subring generated by the ring of rational integers and the element $$i$$ (a square root of -1) in the field of complex numbers.
 * 2) It is the integral extension $$\mathbb{Z}[t]/(t^2 + 1)$$ of the ring of rational integers $$\mathbb{Z}$$, with the image of the indeterminate $$t$$ denoted as $$i$$.
 * 3) It is the ring of integers in the number field $$\mathbb{Q}(i)$$, a quadratic extension of the rationals given as $$\mathbb{Q}[t]/(t^2 + 1)$$ (with the image of $$t$$ denoted $$i$$).