As near as I can tell, this proof would work in any well-ordered integral domain D where D's field of fractions would the role of the rationals. The analogue o

As near as I can tell, this proof would work in any well-ordered integral domain D where D's field of fractions would the role of the rationals. The analogue of the standard proof would require that D also be a unique factorization domain (or maybe the slightly weaker condition that any two elements have a GCD). It might be the case that all these properties together "force" D to be a UFD or that the author snuck another property of the integers in there, but I've only taken a cursory look.