If you took abstract algebra (which presumably you did as a math major), you certainly encountered these at least in the exercises as groups of the form ax + b where x is some irrational number (or imaginary) and a and b are integers are a staple of chapter 1–2 proofs. Gaussian integers (ai + b) are a special case that are loads of fun to play with it. They are not unique factorization domains like the integers (e.g., 5 can be expressed as both 1∙5 and (1 - 2i)² where 1, 5 and 1 - 2i are all irreducible).

Nit: while it is not generally the case that rings of algebraic integers must be unique factorization domains, it is the case for Gaussian integers! In your example, 5 is uniquely factorizable up to units as (1-2i)(1+2i).

Indeed, the integers have the same limitation -- factorization is unique only up to units. 1 = -1 * -1

In elementary mathematics, people wave away "-1" by saying silly things like "positive integers", before Gaussian integers arrive and force us to figure out precisely what we are trying to say without silly ideas from analysis like "ordering". :-)

You are right, of course. Clearly, I have been away from mathematics too long.