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". :-)
