symmetric monoidal (∞,1)-category of spectra
The Gaussian numbers, sometimes (unnecessarily) called the Gaussian rational numbers, are the elements of the number field $\mathbb{Q} + \mathrm{i}\mathbb{Q}$, where $\mathbb{Q}$ is the field of rational numbers. In other words, a complex number is Gaussian (and a fortiori algebraic) iff both its real and imaginary parts are rational. The Gaussian integers are the algebraic integers in the Gaussian numbers, which happen to be simply the elements of the integral domain $\mathbb{Z} + \mathrm{i}\mathbb{Z}$, where $\mathbb{Z}$ is the integral domain of (rational) integers.